Just a quick note about DNA tests, but they're not exactly reliable on the front of being an absolute arbiter of ethnic origin. In other words, you still could have that ancestry, it just doesn't show up. This is due to no design flaw on the part of the DNA test itself, but rather having to do with the way our genetics are passed down.

When sperm and eggs are produced, the genes within are only a part of the respective parent's genome. The genes are mixed and matched as part of what's called genetic recombination, and new combinations are genes end up created while other genes are left out. At the end of this process, you have half a set of new chromosomes in each sperm or egg, that then combine to create a novel full set at conception.

The implication of this process is that the markers used to determine ancestry from a certain area can be lost in the process – they can simply be in the half of the DNA that isn't passed on. These markers can also end up destroyed during genetic recombination, or otherwise lost in mutations in DNA.

On top of all of this, there's a very large issue in that these genetics testing companies don't have access to the DNA of populations from the vast majority of history. So, instead of throwing their hands up and saying "who knows?", the companies use what are called 'living proxies' which basically means they look at the dominant genetic markers of the current groups that comprise said ethnicity. Now, while that's all well and good for those modern groups, the fact is is that these markers can be displaced by migrations, intermarriage, or any number of events, or can simply be lost through centuries of the processes outlined above. So, what may have been a dominant set of genetic markers in 1600's Scotland may now be a small minority, or even lost to the population of Scots in Scotland today.

Finally, I want to end this on the note that genetics often mean very little, if anything, when it comes to identifying members of an ethnicity. Culture, language, personal identification, and acceptance by the ethnicity as a whole mean far more than what one's DNA says (or doesn't). There's a lot to be said about how things like blood quantums, having traceable ancestors on the Dawes lists, and Jim Crow/Jim Crow-esque laws have affected our views on the relationship between what's in our blood versus what everything else about us might say, and these often hand-in-hand with harmful ideas such as racial essentialism. However, I'll leave the details of that to others who can give fuller and more in depth explanations.

There's a lot to this question. In addition to the title question, you've also raised points about the relationship between DNA and claiming (North American) Indigeneity. I want to stress that I am not of an Indigenous group, nor do I study race or racialization specifically, but I am familiar with the current literature in the field due to my advisor and subject area.

First, to your primary point -- although during the 19th century there was a lot of public discourse, discontent, and violence towards Native communities and individuals (looking past the Indian Removal Act, we also have northern removal, the Civil War, and the Californian genocides in the same three decades) , there was social and political power in claiming relationship to Native groups. Chadwick Allen, in his book Blood Narrative, outlines several types of political claims or relationships that people stake out through literature. In responding to Allen, there has been a growth in scholarship concerning 'Settler Indigeneity,' or the process by which non-Indigenous people claim Indigeneity for a myriad of purposes. These purposes include laying claim to ownership or possession over places ( Maile Renee. Arvin, Possessing Polynesians , 2019), invoking a sense of natural belonging or right to live (Lisa Blee and Jean O'Brien, Monumental Mobility, 2019; Shona Jackson, Creole Indigeneity, 2012), or effectually erasing Native claims to the the title of land (Jean O'Brien, Firsting and Lasting, 2010). Many authors in Settler Colonial Studies touch on the idea that, by claiming Indigenous descent, one is able to disassociate oneself from being a settler colonist. (A quick google search will give you plenty of news articles about why people claim Cherokee relationship specifically.)

That's the quick answer. The longer answer, and to respond to the other assumptions in this post, will take a little more time. The relationship between DNA and tribal affiliation is a contentious subject. Blood quantum, or measuring the 'quantity' in fractions of 'Indian blood,' was (and is) a superficial measurement of identity marked by the U.S. Government (prominently during the implementation of the Dawes Act in 1887) . You might ask, "How did the government know what percentage of 'Indian blood' people had?" And the short of it is, they didn't (Naylor, African Cherokees, 2008; Krauthammer, Black Slaves, Indian Masters, 2013; Tiya Miles, Ties that Bind, 2005). Measurements were taken by pricking wrists and seeing what the blood looked like; by taking testimony of family members, but only if they weren't phenotypically Black; by measuring heads and literacy (Lowery, Lumbee Indians in the Jim Crow South, 2010). Blood quantum today is still held as a system of measurement enshrined in some tribal constitutions. But there are plenty of scholars and community members who are very vocally against using blood quantum as a measurement of identity, as blood does not equate to culture (Kim Tallbear, " DNA, Blood, and Racializing the Tribe," 2003. I suggest you check out this article particularly. You can also find a good discussion of race historically here: u/commiespaceinvader Monday Methods: "No but what race were the ancient Egyptians really?" – Race as a concept in history : AskHistorians (reddit.com) .)

Tallbear makes the point that looking at kinship relations and the practice of cultural ceremonies -- speaking language, dancing at gatherings, maintaining connections -- are far more important when determining if an individual is of an Indigenous community than DNA. This conversation is further complicated when speaking about racialization and Blackness, though, as many Black Indigenous people were artificially cut off from the Dawes rolls (Naylor, Miles; in addition, see David Chang, The Color of the Land, 2010; Claudio Saunt, Black, White, and Indian, 2005; and Darnella Davis, Untangling a Red, White, and Black Heritage, 2018). There are many answers on this subreddit dedicated to Cherokee slavery, and I recommend you seek these out if you are further interested in this thread.

Hope this helps. Any errors are my own. *Edited to add username ping for link.

I'd like to add another Kim TallBear recommendation to the list provided by u/MaddestJas: Her book Native American DNA: Tribal Belonging and the False Promise of Genetic Science (2013) is highly recommended for anyone looking into this topic in more detail. For those who can't access the book or her article named in that post, she also speaks on this subject on the podcast All My Relations in the episode "Can a DNA test make me Native American?" and in this Twitter thread.

All My Relations, which is hosted by Dr. Adrienne Keene (Cherokee) and Matika Wilbur (Swinomish and Tulalip), has an excellent two-part discussion of blood quantum in the episodes "Blood Quantum" and "Love in the Time of Blood Quantum". In these episodes, they are joined by molecular biologist Charlotte Logan (Akwesasne Mohawk), attorney Gabe Galanda (Nomlaki and Concow), lawyer Tommy Miller (Colville), and political scientist David E. Wilkins (Lumbee). Their range of expertise mixed with moving personal anecdotes about the effects of blood quantum laws in their day-to-day lives provides an excellent overview of the subject from multiple angles.

Another important factor here is that claiming distant Indigenous ancestry is a way of situating Indigeneity as something firmly in the past. Kim TallBear puts it this way in an interview from 2018:

I spent a lot of time in Charlottesville, Virginia, and there are all these white people around there that love to claim they are descendants from Pocahontas—privileged white people who have no lived relationships with indigenous people. To them, indigenous people are back in the 1600s, are back in the 1700s. We don’t really exist for them as living relations today. It really is, Well they’re dead and gone, so we can just claim to be them. There’s none of them around to correct us anyway. They really do believe, I think, in the East that we’re largely dead and gone. That’s what I discovered when I moved to Boston from South Dakota back in 1989. It was just a shock to me how many [...] good-meaning progressives said, Oh, I thought there were no natives left. Can you guys leave the reservation? They asked me the craziest questions and showed me they had no reference points for native people at all, and so that’s the kind of environment that somebody like Warren I think is operating in.

By claiming Indigenous ancestry, then, many white people, even if they do not consciously realise it, are participating in a cultural process which was set into motion in the 19th century: the idea of the "vanishing Indian", or assimilating Natives to white culture so effectively that Native Americans would cease to exist. This is the ultimate project of settler colonialism, as laid out in some of the sources listed in u/MaddestJas's post. Blood quantum was designed with this goal in mind as well, since the idea was that if you defined Native identity by a percentage of Native blood, intermarriage with whites would eventually erode that percentage to nothing.

To explain how closely linked settler land claims and the goals of Indigenous erasure are to the persistent myth of the distant Native American ancestor, I'm going to look at one case in detail: US Senator Elizabeth Warren. Warren forms the most famous recent case study of this phenomenon. She grew up in Oklahoma, where claims of Indigenous heritage are common among white people. When Harvard Law School first hired her in the 1990s, she was promoted as a minority hire, and Harvard highlighted her as the first minority woman given tenure. She listed herself as a minority in the Association of American Law Schools directory, a change from when she had listed herself as white in the 1980s in the personnel records of UT Austin, and one which she repeated at the University of Pennsylvania in 1994. After receiving tenure at Harvard in 1995, she went back to listing her ethnicity as white, though Harvard continued advertising her as a prominent "woman of colour" afterwards.

This issue was first highlighted by her political opponents during her Senate run in 2012. Then, as part of a back-and-forth with political opponent Donald Trump, she later took a DNA test to meet his challenge of "proving" that she had Native American heritage. The DNA test claimed that she had between 1/64th and 1/1024th "Native American heritage". Warren faced serious backlash from scholars and other voices across Indian Country over using this result, as well as racial stereotypes such as "high cheekbones", to defend her claim to Cherokee family heritage. Although she has since walked back many of her claims, diving into her family history exposes how claims like this came to be.

Cherokee genealogist Twila Barnes, who goes by the online moniker Polly's Granddaughter, has conducted in-depth research into Warren's family history. Although Warren claims that her mother was discriminated against because of her Native American ancestry, her mother Pauline Louise Reed is listed on all US census records as white. Both of Pauline's parents are also consistently listed as white in census records going back to 1880, and their parents are consistently listed as white going back to 1860. Warren's great-great-grandfather, John Houston Crawford, did live in the Cherokee Nation in 1900. Unlike his Cherokee neighbours, however, he was listed as white, and he did not appear in the Special Schedule for Indians census from the same year. He, nor any of Warren's other great-grandparents, does not appear on the Cherokee Census or the Cherokee rolls. The same is true of her great-great-grandparents, who are all listed as white and completely absent from Cherokee records.

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