Now this is a really interesting question, and I'm going to preface my response by saying that there's no really clear-cut (or entirely comprehensive) answer, because your chances of recovery will be largely dependent on time, place, and, as you have pointed out, your particular injury or disease. My area of study is European medicine, so that is what I will be able to discuss with the most authority. If anybody else has examples of Middle Eastern, East Asian, African, or Native American treatments, I would love to hear them. Also, bear with me, because I cannot but answer in a somewhat roundabout way.

The first thing you should know is that the history of medicine has been largely influenced by anthropology, in that it tends to consider illness by taking into account its cultural specificity. Obviously, different cultures understand diseases in different ways, and have different rituals and psychological experiences surrounding what we may consider to be the "same" disease in biomedical terms. Furthermore, it is now known that people from different societies can actually have different bodily reactions to the same pathogen. In other words, your illness experience is largely shaped by your background, not just in terms of lived experience, but in terms of how you physically experience a disease.

Now, let's take a moment to consider the placebo effect, "the well-documented fact that the bodies of people who think they are receiving effective treatment actually respond positively, even when the treatment is no better than standard treatment, and even when the treatment is not a treatment at all." The placebo effect works two ways: on the one hand, it can convince people (and their bodies) that they are receiving treatment when they actually are not; on the other hand, people (and bodies) who do not believe they are receiving effective treatment may not respond to real drugs. (Edit: Changed "will not respond" to "may not respond," in recognition of criticism that I have overstated the placebo effect. I merely intended to point out that people may respond to treatment according to their belief in that treatment's efficacy.)

How does this relate to your question? Well, for one, it means that the example you are proposing - of a person going back in time from the present - is somewhat moot. Let's say you were to head back to thirteenth-century Salerno (modern-day southwestern Italy), and seek treatment for a non-specific disease from a physician who had graduated from the prestigious medical school there. You are diagnosed with "tertian fever" (note that fever is its own diagnosis at this point, not a symptom), which is a kind of fever known as "putrid" - meaning that it affects the humours. In this case, the affected humour is your "red bile," which the doctor tells you has putrified because your body is too hot and dry. The doctor will try to cure you through a combination of regimen (frequent cool baths, cold and moist foods like lettuce), drug therapy (especially purgatives), and perhaps even bloodletting. Most of this would likely seem preposterous to you - "What do you mean I have a 'fever', that's not even a real disease!" Your condition, whatever it might be by modern standards, would probably not improve, regardless of the "objective" efficacy of the treatment, because you do not believe that the treatment will work. So in essence, the answer to the question "has medicine always been useful?" is an unequivocal YES. However, the same medicine that worked for a thirteenth-century Salernitan would not necessarily have worked for YOU, were you to go back in time.

Now that I've dispensed with the obligatory "history of medicine" disclaimer, I can attempt to give you a slightly more satisfying answer. As you have noted, your efficacy of treatment by modern-day standards will be very relational to your diagnosis. A broken bone could be set just as effectively in the third century as in the twentieth, whereas something like cancer would not only be treated in very different ways, but would be understood in mutually unintelligible terms. I'm not sure whether there's a definite point during which you would be risking your health to see a doctor. Prior to the advent of biomedicine, medical services were actually very patient-driven, and it's unlikely that you would have been forced into a treatment you saw as harmful against your will. Indeed, many forms of therapy we might see as harmful were also distrusted by university-trained doctors. The most prominent example I can think of is "couching," a surgical treatment for cataracts that often caused permanent blindness. "Professional" surgeons (such as they were) portrayed themselves almost as if they had been manipulated into performing these procedures; on the one hand, they recognized the danger of the operation, but on the other, they knew that if they didn't carry it out, the patient would simply go to a less qualified or less scrupulous surgeon, who ran a higher risk of blinding them.

That being said, there are some specifically pre-modern treatments that could be considered effective by modern standards. About a year ago, in a rather high-profile finding, historians and microbiologists from the University of Nottingham discovered that a particular Anglo-Saxon remedy from the ninth century, when prepared exactly to specifications, could cure an antibiotic-resistant form of eye infection called MRSA.

But throughout all of this lengthy explanation, I suspect your question is actually closer to "How far would I have to go back before medicine was no longer broadly effective for me," in which case we need to identify the point at which medicine stops being recognizable as medicine for you. That point, in all likelihood, is the mid-nineteenth century, which is the point at which the theory of modern biomedicine really came to be, along with its symbols, treatments, and rituals. From the 1850s to the 1910s, most of the things you would probably associate with "good" medicine became a part of regular practice - including anaesthesia, aseptics (i.e. hand-washing and sterilization of instruments, which were understood through germ theory), and widespread vaccination. The doctor's office, the hospital, and the operating room would also be uniquely recognizable to you, as would be the techniques of diagnosis and the type of communication between you and your doctor. Although a doctor, at this point, might still propose a diagnosis or treatment that seemed absurd to you (e.g. neurasthenia treated by shock therapy), they would be able to explain it in a way that made sense according to your modern understanding of disease, at least in most cases.

There are all kinds of disclaimers to add to this explanation. I'm mostly dealing with mainstream "professional" medicine in this response, and not so much taking into account medicine without a coherent explanatory framework, or the experience of consulting a modern practitioner of alternative medicine. Furthermore, as you have identified, there are modern treatments that have only been relatively recently considered harmful. Smoking is a good example of this, as is lobotomy, which actually made a lot of sense as a treatment within a modern explanatory framework of mental illness. Indeed, some types of lobotomy are being resurrected in surgery, although not the pre-frontal lobotomy that was popular in the 1940s and 50s. On the whole, this is a very difficult question, but a very interesting one, because the answer can be approached from all kinds of different perspectives.

SOURCES:

Placebo effect, and its impact on understandings of historical disease:

• Faith Wallis, ed., "Introduction," Medieval Medicine: A Reader (Toronto: University of Toronto Press, 2010).
• Faith Wallis, "Medieval Medicine: Introduction and Overview." This is unpublished, but I'd be happy to send relevant quotes on the question of efficacy in medieval medicine to anybody who is interested. The author is a very well-respected medievalist historian of medicine.

Cultural relativism in medicine, more generally:

• Anne Fadiman, The Spirit Catches You and You Fall Down (New York, Noonday Press, 1998). This is an amazing work of medical anthropology, and I'm always looking for chances to recommend it just as a piece of non-fiction as well, because it's a really excellent book.

Medieval treatments:

• Michael McVaugh, "Cataracts and Hernias: Aspects of Surgical Practice in the Fourteenth Century," Medical History 45 (2001): 319-340.
• trans. Faith Wallis from the edition by Gregor Maurach, "Johannicius. Isagoge ad Techne Galieni," Sudhoffs Archiv 62.2 (1978): 148-174. Latin.

MRSA, including the "recipe" from Bald's Leechbook:

• https://www.nottingham.ac.uk/news/pressreleases/2015/march/ancientbiotics---a-medieval-remedy-for-modern-day-superbugs.aspx

The rise of modern medicine (if you want to try and pinpoint a little bit more specifically the furthest back you, personally, would like to go back in time when it comes to medical treatment)

• Deborah Brunton, Medicine Transformed: Health, Disease, and Society in Europe, 1800-1930 (New York: Manchester University Press, 2004). I would especially recommend "The Localization of Disease" (L.S. Jacyna, 1-30), "The Emergence of Modern Surgery" (Thomas Schlich, 61-91), and "The Rise of Laboratory Medicine" (Deborah Brunton, 92-118).

In the USA:

• If you suffered from mental illness, medical intervention was bleak until the mid 1950s. Asylums were generally atrocious, but I'll focus on one aspect: The first prefrontal lobotomy on American soil was performed at the George Washington University Hospital on 14 September 1936 by the neuropsychiatrist Walter Freeman and his friend and colleague, the neurosurgeon, James W. Watts. By 1951, almost 20,000 lobotomies had been performed in the United States.

Ref: Pressman, Jack D.; Last Resort: Psychosurgery and the Limits of Medicine. Cambridge: Cambridge University Press; 2002. ISBN 0-521-52459-8.

Rose Marie Kennedy, John F. Kennedy's eldest sister, displayed behavioral problems. Her father arranged one of the first prefrontal lobotomies for her at age 23, but it failed and left her permanently incapacitated. She spent the rest of her life in an institution in Jefferson, Wisconsin, with minimal contact from her family.

• If you were mentally retarded, medical intervention was bleak until the 1960s: The 1927 Supreme Court case Buck v. Bell which legitimized the forced sterilization of patients at a Virginia home for the mentally retarded. About 60,000 forced sterilizations occurred nationwide from 1909 up until the 1960s.

Ref: Stern; Eugenic Nation: Faults and Frontiers of Better Breeding in Modern America. University of California Press. 2005

• If you were a poor black farmer, medical intervention was risky until 1970s: From Wikipedia: The Tuskegee Syphilis Experiment was a clinical study conducted between 1932 and 1972 by the U.S. Public Health Service studying the natural progression of untreated syphilis in rural African-American men in Alabama under the auspices of receiving free health care from the United States government. ... After funding for treatment was lost, the study was continued without informing the men they would never be treated. None of the men infected were ever told they had the disease, and none were treated with penicillin even after the antibiotic became proven for the treatment of syphilis.

In 1966 Peter Buxtun, a PHS venereal-disease investigator in San Francisco, sent a letter to the national director of the Division of Venereal Diseases to express his concerns about the ethics and morality of the extended Tuskegee Study. The Center for Disease Control (CDC), which by then controlled the study, reaffirmed the need to continue the study until completion; i.e., until all subjects had died and been autopsied. To bolster its position, the CDC received unequivocal support for the continuation of the study, both from local chapters of the National Medical Association (representing African-American physicians) and the American Medical Association (AMA).

Current answers seem to focus on how things we take for granted weren't in a frighteningly close past. I know that archeology showed that people have survived trepanning in a surprisingly far past. Do we know if that was done for medically sound reasons? Are there other similar examples?

And I remember reading about a man ridiculed for suggesting doctors should wash their hands...

I can't speak to the rest of your question, but I thought I'd expand on this, as Ignace Semmelweis is a hero of mine.

Basically he investigated deaths from "child-bed fever" (infection after giving birth) at a maternity hospital in Vienna. The hospital had one wing for women wealthy enough to hire doctors, and another for indigent women - who were attended by midwives.

Contrary to what might have been expected, women were at a significantly higher risk for infection if they were attended by doctors. Semmelweis noted one difference; that the men (women were not yet admitted to med school) attended autopsies. Whereas the midwives did not. He posited that there was some agent of infection that the men were carrying to the labouring mothers.

He was able to show that washing hands in a solution of chlorinated lime reduced fever rates dramatically, but this was before germ-theory was really developed, and Semmelweis could not convince the doctors to follow even simple hand-washing procedures because they refused to believe in something they couldn't see.

Follow-up question: Would I have been in trouble if Galen had treated me for some arbitrary illness/condition?

I was under the impression that he was performing some rather sophisticated surgeries, the thing that comes to mind (true or misremembered) was that he performed cataract removals rather well. Wasn't he setting broken bones correctly, stitching lacerations and so forth?