Thanks for re-posting this Martin. It deserves a thread to itself.
It is an interesting presentation, but he contradicts himself. At 4.20 he says "about three quarters of our six great grandparents we don't share any DNA with". That suggests we only share DNA with about 64 people out of the 256 ancestors at that generation. But later on, with his simulation from about 5 minutes in, the chart shows that we could actually share DNA with up to 93 out of our 256 six times great grandparents (which is over a third). A small point, but in mathematics a small variant can throw out a huge error in the long run. His math also doesn't allow for any cousin marriages (a huge missing factor in my opinion) which will throw his conclusions way out.
But I get the point. Basically, DNA can only prove part of our ancestry, since we don't inherit DNA from every one of our ancestors. Therefore ethnicity estimates are only based on the DNA markers we have inherited from a certain number of our ancestors (i.e. not all our ancestors). So even if the ethnicity estimates were accurate it would ignore a huge percentage of our ancestors whose DNA we have not inherited.
Therefore if I get told I'm 79% Irish/Scots and 21% North European, yet I believe that one of my 3 x great grandparents was Indian, and another was North African, I can ignore the estimate and go with my family knowledge (if I have document proof, of course). But the danger is if i didn't know about those ancestors, and believed the ethnicity estimates were accurate, then I might ignore possible future leads that would discover those Indian or African roots, because I would think "that can't be true because my ethnicity estimate shows I don't have any DNA from there"!