Hey, a question that I know how to answer. Isn't that nice? For reference my answer will relate mostly to England between 1720 and 1820, although I might stray outside there at some points. The long story short is that if you got a disease, it was very unlikely that a doctor or anyone else would be able to cure you. As such the focus of medicine was on preventing illness rather than curing it. In the dominant medical theory of the time the environment was the chief cause of disease -- particularly 'bad air' and dirtiness. So you have people like Wesley (the founder of Methodism/medical writer) arguing for what he calls a 'regimen' or temperate lifestyle which he thought was conducive to health. Others like Buchan (author of a very popular treatise on domestic medicine) arguing felt that "Unwholesome air is a very common cause of diseases... Air, as well as water, takes up parts of most bodies with which it comes into contact, and is often so replenished with those of a noxious quality, as to occasion immediate death." The argument was that if people lived a temperate lifestyle, ate the right food and lived in an airy environment then all would be well.

However, when people did get sick there were certain things that they could do. In most circumstances, if you got a serious condition like tuberculosis, yellow fever and so on, you were likely to die, and there was very little that a physician or unlicensed quack could do for you. For a very select group of conditions there was some means of surgical relief. Broken bones could be set by a barber surgeon, cataracts could be removed with a reasonable degree of success, as could hemorrhoids.

However, this still leaves a large section of debilitating conditions which affected peoples lives tremendously while not killing them -- things like asthma, gout, rheumatism and sexually transmitted infections. We can get a flavour of how this affected people by looking at the old editions of the Gentlemen's Magazine. It was a sort of eighteenth century version of GQ and the the most part reflected the views of the educated, enlightened elite, mostly based in London. What is interesting about it for our purposes is that it included a large medical section in every issue. These people loved to self diagnose and discuss their ailments. Other correspondents would then write into the next edition, copying out sections from their own medical books, or positing solutions which had worked for them in the past. Solutions include honeyed water as a way of helping with Asthma, or various wonder cures like Bishop Berkeley's tar water or the (poisonous and made with Antimony) Drop and Pill of Dr Ward.

So a short and provisional answer to your question would be this. Was everyone sick all the time -- no. Many people enjoyed good health their entire lives. There was however a large portion of the population which was afflicted with a whole host of conditions which were beyond the medical technology of the time to cure. They had to live with it and cope however they could. As to having more than one disease at a time, yes that was very much possible. One example which comes to mind is the Parisian physician Laennec. He suffered greatly from the tuberculosis which eventually killed him, alongside asthma and various other lung and bowel complaints. Another surgeon, John Hunter, managed to infect himself with both Syphilis and Gonorrhoea as a result of an experiment gone wrong. While these cases are of course to a certain extent unrepresentative, it would certainly not have been uncommon for people to have been picked up a variety of parasites over the course of their lives alongside other ailments due to age.