I have a cousin who has been working on our family tree for most of his life and he believes that we are descended from Edward Longshanks (Edward I, Plantagenet). I don't think so, but he is convinced.
Why not? When a monarch has daughters, like Edward I had, who marry into the nobility, and their offspring continue the downward social spiral by sinking or marrying into the gentry, then the yeoman class etc. it doesn't take many generations to reach somebody as humble as you and yours truly!
With regard to medieval monarchs, the descendants had to stay within the armigerous classes untill keeping of records of baptisms, weddings and deaths also was extended to average folks (some time after the Reformation or Counter-Reformation, depending on area), otherwise it's impossible to document.
But recent examples of how quick social descent is possible are e.g. (morganatic) Romanov scions and The Lady Iris Mountbatten, great grand-daughter of QV, and her descendants. And consider this: QEII, the current high priestess of royalty, can trace her ancestry on both her father's and her mother's side back to the Dark Ages through many lines (if we keep undocumented mythical descent from King David, Wodan etc. out of the picture!), but one maternal line stops in the last half of the 18th century, with a certain Frances Webb (the Queen Mother's maternal great great grandmother). Who was this Frances Webb? And how removed was she from the lowly weaver who probably was the source of her surname? Or consider hereditary peers called Cooper, Le Tonnelier de Breteuil etc. That means one of their ancestors was a mere cooper!
If QEII can be descended from an obscure 18th century woman we know nothing about, I can't see why you can't be descended from a 13th century monarch! If you give it a few generations and throw in lots of hard work, sex drive, crime and disasters, then social mobility, in both directions, can be amazing, not only in the US, but also in the ancien-régime Old World.
