> far more children are not the children of their named or assumed father than was suspected

The origin of this factoid is one British study about paternity tests for fathers on some street. However, the methodology of that study was deeply flawed and the fathers self-selected (i.e. they already suspected they weren't the biological fathers). Later studies has put the rate of "false fatherhoods" at about 1%. https://www.iflscience.com/false-paternity-isnt-actually-wid...

I've never heard of the British study you propose being the source of my observation. But it doesn't matter. If we're talking about deep ancestry, like say tracing your lineage back to Charlemagne (which seems to be a popular anchor), you're talking about 40 to 50 "fathers" between the good old King and a modern descendent. If there is a 1% probability in each generation that the paternity recorded was wrong, then you're looking at at least a 1 in 3 chance that your paternity chain is broken. If the rate is even as high as 2%, it's 1 in 3 that the chain isn't broken. One break, of course, and your claim of descent is invalid.

Matrilineal or patrilineal lines give you much stronger signals than that, though, thanks to mitochondrial and Y-chromosome DNA respectively.

I don't see how. Sure, if you had mitochondrial DNA from your long-dead male ancestors, you could suss out your true patrilineal line of descent, but it's extremely unlikely that such data is available. You might be able to get it by inference by analyzing the fanned-out descendents of a relatively recent putative paternal ancestor - back 8 or maybe ten generations, but even that's unusual. And, if you're targeting a specific ancestor, it requires an unbroken line of paternal descent - as soon as your path to, say, Henry VIII goes through a female descendent, at any generation, you've lost the Y chromosome relationship. Same argument applies with the sex roles reversed for females and mitchondrial inheritance. At ten generations back, you've got 1024 grandmother leaves on your tree. Only one of them is your mother's mother's .. mother's mother. At 20 generations (say, 500 years, so the actual age of Henry VIII), you've got a million grandmother or grandfather leaves on your tree, and only one is purely matrilineal, and one purely patrilineal. (That doesn't mean, of course, that you've got a million distinct grand^20 fathers - it'll be way less than that, because the some men will appear on many different leaves, and the same for mothers. But the odds that the path you claim to Henry VII is purely maternal or purely paternal are still very small.

Sure, but unbroken patrilineal or matrilineal lines of descent are rather improbable over such long times - any family that had only sons or only daughters breaks the line, and that's not particularly uncommon. Add some infidelity into the mix and it becomes even harder to believe.

Non-genetic descent is actually much stronger than that even.

When genes are sexually recombined, they don't recombine randomly per-base pair, they're cut and recombined in big strips. The chromosomes actually only 'snip' in a small number of places.

This means that after a relatively small number of generations, it's quite easy to have no genes at all from one of your ancestors.

I don't recall the exact numbers but it's something like 8-12 generations where you start having serious numbers of ancestors with no genetic connection.

Of course. Absent recombination, you'd start having ancestors dropping out after only 5 generations. With recombination, it's a few more. I didn't make the argument that way, because when I have in the past, I get a bunch of "yeah, but ..." rebuttal. In addition to the complicated nature of recombination, there is the additional complication that when you get back 6 to 8 or more generations, the same progenitors generally start appearing in multiple branches of your family tree - sometimes in many, many branches. But it is nevertheless the most correct. We all probably have great, great, great, great, great, great grandparents from whom we have no genetic inheritance at all. Add a couple more greats, and it's a near certainty.