It depends on the area and how early they had writing and what academic tradition is being followed, generally speaking. In my case (Japanese history), we begin the "historical" period in the 6th/7th century CE since that's when there are actual written records - genealogies, law codes, and receipts (mokkan) mostly. Earlier than this tends to be more the domain of either archaeology (in the US this is housed in Anthropology rather than History) and Art History. There are some Chinese records that talk about Japan from the 3rd century, so between the 3rd-5th centuries is usually referred to as "protohistoric." Historians can study it, but there are no primary written documents from Japan.

That's not to say that historians can't study a place or people without written sources of their own. It's possible to read other's writings about the place (like the Chinese Wei Zhi from the 3rd century that talks about Japan) and use material culture, but that's not what we're trained for. We're trained to use the written word as our primary means of study. Anywhere else, like China or the Levant, will be much earlier since archaeologists have recovered writings on things like clay tablets in the Levant and tortoise shells and ox bones in China. In that case, you can go back thousands of years.

Thats an interesting question and one that I think has a short answer of depends, with a longer answer below.

When you're looking at history, you're looking at a written or physical evidence trail of an event or culture, with the notion that it fits into some narrative or argument. So in that regard, a historian that looks at history with an eye for interdisciplinary studies (including anthropology, archaeology, geology, etc) can have their range extended back beyond written evidence, depending on what they want to do with the evidence, and what these interdisciplinary studies show. For example, if I want to look at pre-Rome Italy, I'm going to have to rely pretty heavily on non-textual sources due to climate and just the lack of strong written records. But that means that I'm going to go for much more of a "broad strokes" picture, as opposed to something more specific. Continuing on this example, I could say that votive sacrifices and offerings were made in central sacrificial pits, but I can't say that they made sacrifices every other Tuesday at 6 in the evening. The archaeology bears out the presence of these items as well as depictions in art, but not the frequency. Additionally, you can use the material record to show what kind of connections are present within a non-writing society (the example of Chinese silk being found in a Nordic grave in Ireland and the prevalence of Great Lakes copper among Native Americans come to mind),but again this is just to create more of a base understanding of a culture or society. The same thing goes with the study of remains and detritus that can be found at sites.

But this also poses another interesting issue that I think books like Peace Came in the Form of a Woman (Juliana Barr) try to handle; what do we do about oral traditions. In the past these have been discounted as inconsistent and non-reliable, but at least from my own perspective in graduate school, this has been changing over the course of the last decade or so. While they can't give specific dates, oral traditions or carvings can give us a great look into the past of a culture, while still eluding us in neat and tidy dates that we want. In some instances we can try and backdate (if an event was said to happen on a solar or lunar eclipse) but in others, we can't really. In her work New England Bound Wendy Warren uses the bodies of enslaved Africans to help tell their story of capture and enslavement in a way that's both powerful and in my own opinion, interesting. So in that respect, we do see that scholarship is looking at the oral traditions and even the bodies of individuals involved in history in a way that you might not have seen at the turn of the millennia.

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Additionally, you can look at sources that might extend further back then the written record of your subject. And in some cases that can help. Classical sources describe the Germans (in an albeit biased and pretty awful way) before strong written records for their cultures really exist. So we can get a somewhat lateral picture of them, but only in the way that these historians or authors viewed them. That can create issue, but it does allow for you to see how contemporary cultures view a subject that either doesn't have a strong written record, or has none at all (or at least not one we can currently translate). This can create its own issues though, and is worth reminding again to take it with a few pinches of salt more than a margarita, if you'll pardon the expression.

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I come from a background talking about borders and frontiers in the medieval and early modern world, so I'm usually comfortably within the realm of having texts available. But it isn't always what you need. Primary sources do create the bedrock of good history, but interdisciplinary sources if used correctly and not over-estimating what these sources "say" can allow you to extend beyond the range of what texts may use. So while the easy answer is "as long as there's writing," I think the more nuanced answer (that others here can attest to) would be that it would depend on the prevalence and reliability of sources that allow you to create what I would deem a reasonable understanding of your topic.

The Early Dynastic Period of Sumerian History and/or whatever the very early Egyptian equivalent would be. The debate between which civilization invented writing first has gone back and forth for a long time. But it is certainly either of these two civilizations at around 3000 BC where texts begin to be used in our study of these cultures. So that's where I'll plant my flag for what is the earliest period for a "Historian" to study.

For an archaeologist or anthropologist, then the limit is as far back as hominids have existed...