Different tribe Dutch.Europeans tended ( and in most parts of Europe still do ) to not mix between tribes.
Completely untrue. There has always been a huge deal of migration and continuous movement of people all across Europe since human beings first came there.
If you take the average European and look back five generations you're probably finding ancestors of all ethnicities in Europe and many from beyond.
Just by looking at surnames I could tell you that among my friends there are people from all over the place - and that's the case even though poor families usually didn't have a surname a few centuries ago, which actually dilutes the whole picture in favour of your thesis.
Also, it's realtively unbelievable, that Van Buren is not related to any English king. Or that I'm not, for what it's worth. Just because she couldn't find any relation between the two of them doesn't mean they are not related.
Btw, there have been very interesting genetic tests in regards to Genghis Khan:
http://en.wikipedia.org/wiki/Descent_from_Genghis_Khan#DNA_evidence_-_The_Genghis_Khan_Effect
http://news.nationalgeographic.com/news/2003/02/0214_030214_genghis.html
8% of all men living in the area of the former Mongol empire are direct descents of Genghis Khan or close family members of his, that's 0.5% of all male population in the world. It's not that unbelievable to assume almost everyone in this region is in one way or another related to him. And probably, if you go back in time long enough, everyone on this board too.
Also keep in mind that in 1000 AD there were only ~250 million people in the whole world (30 million in Europe). In 1500 the number was probably slightly over 400 million, and 70 million in Europe only. Because of the industrial revolution human population exploded and grew exponentially, but with a relatively small number of ancestors.