The answer is most likely yes, but not just for the presidents. This article by The Guardian explains it best. But essentially, the further back in time you go, the higher percentage of that population can be traced to you.

It's scientific fact that you have a biological mother and father which leads to 4 total grandparents, 8 great-grandparents, etc. By the time you go back 25 generations (which would roughly take you back to the time of King John), the math says that there were 528 million people who directly led to you (and this doesn't even include aunts and uncles). However, in reality, it works out a little differently. Obviously, there weren't 528 million people in all of Europe back then, so at a certain point, the family tree looks less like a tree and more like a giant web. There's a lot of interconnecting bloodlines that had already been connected at some point previously.

The Guardian claims that nearly everyone on earth is related to Charlemagne (who lived in the 800's). Therefore, quite a large number of people (especially those with European descent like our Presidents) can be traced back to King John as well.