Interesting, is though, through my own and other's informal research, I have found 100's of ties back to Welsh, English, and Scottish Royalty.
Seemed far fetched, until I realized that this is probably the case for most of us. The poor prior to the 1700's were continuous fodder for wars, disease, starvation, etc. The non-direct kin of nobility filtered down into the "middle class" continuously, in a sad but true form of natural selection.
With that said, getting into the Plantagenet Society is not easy to get into to. One of the best known genealogist in the country G. B. Roberts has what is considered to be the reference guide to those who are eligible. Roberts acknowledges that there are significant amount of people who have a Royal Blood Line, but as in any other heraldic society, proof is needed. In lies the problem. Roberts states that there are only 600 ancestral American immigrants that have the absolute documentation of proof. (The list does grow slowly has his original guide was RD 500, and now RD 600) And in my case, only 3 of the 600 are ancestors of mine.
I am not a PSmember, as SAR is enough for me at this point.
Most Brits who are interested in history/genealogy blame the Black Death for making everyone in England related to some degree. I notice when ever I watch British tv/films, how often people look alike or remind me of other people. My husband jokes that the English have about ten faces.