I'm going start off with this: Any comments about Warren's politics will be deleted.
Elizabeth Warren has, for years, claimed to be an American Indian. I have no doubt she believed it to be true and now DNA testing has validated the claim - to a point. I doubt she can meet the blood quantum of any recognized tribe. My point here is that Elizabeth Warren is not alone. There are probably more of us claiming "great-grandpa was an Indian princess" than there are doing the opposite. (That line in quotes? I totally stole it from a book called How To Talk Trash In Cherokee.)
Like Elizabeth Warren, I've grown up with stories of American Indian ancestry. I've never claimed to be anything but the white girl I am, because I've got no documentation and because it's so far back I've probably got like one chromosome. Most of the tales agree it came down the Parker line, and photographs of my mother and her father enforce that theory.
All the documentation I've found on my genealogy, however, lists the race as white. This isn't proof that we're "lying" about the DNA being there. Many American Indians chose to pass as other races, largely to avoid things like The Trail Of Tears. Some claimed to be black, which speaks volumes about how they were treated... they'd prefer to be seen as black even before the Civil War! I suspect something like this happened in my family.
Why do I insist on still believing it? Because there is other evidence. Physical Anthropology recognizes three skeletal types - one of which is Asian/American Indian. Some of the physical markers? High arches in the foot -- My mother had trouble finding shoes that fit comfortably. (Not that she wore them often, but I digress.) Winged incisors and shovel-shaped teeth. High cheekbones, of course. Shape of the eye socket.
With the skin on, of course, you also get the skin color. All of these things are present in my documented-white family, varying in degree by individual. Other physical indicators I've heard, but honestly have not deeply researched: second toe being longer and slightly separated from the big toe. Tendency toward diabetes and thyroid issues. All present in my family.
A note about DNA tests: DNA can "fall away". If Elizabeth Warren's test had not shown her native ancestor, it's possible he might still exist, especially considering the number of ensuing generations. There is also the possibility that said ancestor was adopted into the tribe and therefore not genetically a part of said tribe.
When all the fuss started about Elizabeth Warren lying about her heritage, I was insulted. She believed it to be true. Just like me. If the university in question took her at her word, not requiring documentation, that's hardly her fault. To say she lied is to say every one of us seeking our own hiding ancestry is a liar.