I work in Indian Country (this is a term of art generally accepted by tribes and tribal members). Most of my co-workers are tribal members, of one tribe or another. I used to practice Indian law. Although it's been a long time, and I've forgotten most of what I used to know. Many of my good friends grew up on a reservation. A couple of thoughts:
1. The laws related to tribes, tribal members, and the relationship between them and the U.S. government is incredibly complex. Lots of rules don't make sense.
2. There is a long and painful history with respect to the Trust relationship the government has with Tribes and Tribal members. For example, many folks in Indian Country feel strongly that the U.S. MUST honor all of the trust duties that it has taken on (including, but not limited to, treaty obligations). Many of the trust duties don't actually show up in treaties, but are a result of a complicated interpretation of common law trust relationships. In short: Tribes want autonomy and self determination and sovereignty, but they also want the U.S. to continue providing the support it pledged 100 years ago. Sometimes these concepts are at odds.
The above doesn't really have much to do with the Warren situation. What does is:
3. The U.S. government does not consider you a federally recognized member of a federally recognized Tribe unless the tribe so recognizes you. In very simplified terms: you aren't "Indian" in the government's eyes unless you are a member of a federally recognized tribe (there are some weird things going on where a tribe is recognized by a state, but not the feds, such as the Lumbee in N.C., but we'll leave that aside for now).
4. The reason for the above, essentially, is that being a member of a tribe is not about "race," but rather "citizenship." The idea of taking a DNA test to "prove" your citizenship into a tribe is almost laughable.
5. Most tribes have some sort of "blood quantum" requirement. In those tribes, you need to be able to trace your lineage directly to someone on the original tribal rolls. I've had friends who have had crappy situations because they may be 1/2 Tribe A, 1/4 Tribe B, and 1/4 Tribe C, and there spouse would be similarly situated. With one of my friends, her kids can be enrolled members of her tribe because they don't have the adequate "blood quantum" for that Tribe, even thought the kids themselves may be like 75% "Indian blood." It's just a weird situation.
6. All of this is to say that, I'm a HUGE Warren fan, in terms of policy, gumption, feistyness, everything. But I'm kind of mad about her claiming Indian blood. Should it disqualify her for running for president? I don't know. Maybe. Because it's stupid. As a country we have crapped on Indians for the last 400 years. If my great great great grandmother was from Africa, but I was raised as a white kid from the suburbs, I'd feel stupid listing "black" on my website somewhere. In fact, my kid sisters have a different father than me, and their father was clearly considered black or African American. But they look white and were raised as white kids. They wouldn't think of trying to call themselves black on a job application.
7. TL;DR -- As much as I love Warren, I think she kind of sucks for identifying as part Native American, especially if it was based on "family lore."