Henry the VII is the main breakage that people point to as being quite suspicious - he was only distantly related to Richard III (of Shakespeare fame), who was killed in battle with him. His claim was extremely tenuous to say the least. During the War of the Roses many other people who had a claim far better than him were killed, but since Henry was the last man standing he won the day.

There were rumors that within his line that either he himself was illegitimate or one of his ancestors that his claim relied on was. There's no real evidence to indicate this was the case, but then again it's something that we're looking at more than 500 years ago that people would go out of their way to hush up. It's not something we'd see in the historical record, especially since the guy that wouldn't want it to get out the most was the king.

Regardless, the sensitive nature of any breaking point makes it unlikely to be incontrovertible, simply because there's a lot of people wishing to muddy the water so to speak.

As a follow up, how far back is DNA analysis possible--as in, how far back is the last known grave?

By way of a follow-up, does anyone with knowledge of the subject have an opinion on Tony Robinson's Britain's Real Monarch documentary, which asserts that Edward IV was illegitimate?

Can the citizens demand DNA tests on the Royal Family? If they are supported by taxes due to their blood line, it seems logical to test them for the possibility of fraud.

Of course there's been adultery - that's why you'll see people sometimes described as "natural" children of a monarch. Henry VIII's "natural" son, Henry FitzRoy, was born out of wedlock, but Hal was fond of the boy and made sure he was taken care of. However, it was incredibly important for queen consorts to make sure that there could be no stain of adultery on any children they may carry... the ultimate goal would be to see your child take the throne from his father, and if he'd been conceived out of wedlock, people would know.