I asked the great marine biologist Roger Payne about this recently because my Ancestry.com test showed no Indian DNA. I have masses of testimony about Indian blood in the form of affidavits in the Dawes Commission files as well as testimony of dark older relatives on my father's side. We were part Cherokee and probably more Choctaw, but after enough generations it can run out. I will bet my dark oldest brother, long dead, kept a little of the Indian DNA, but I got the blonde side too many generations in a row. It happens, and it does not mean that my Indian heritage is any less real. I mean, my great aunts as late as 1990 were talking about their "Chockie" grandmother in some detail. I did not get enough to show in my DNA, but my Indian heritage is real, like Elizabeth Warren's. Silly Scott Brown, who wants to move back to Massachusetts so he can run for the next vacant Senate seat, is ridiculous in demanding that Warren show him her DNA test. I guess anyhow who has shown us his naked pictures thinks there are no boundaries. Now, he did look good in those ads, but lay of Liz, and stay in whatever that state was that you moved to and lost in.
If your great-great-grandmother was ¼ Cherokee, then it was her grandparent that was 100% Native American. And that would be your 4th-great-grandparent. Now your great-great-grandmother would get 50% of her DNA from her mother and 50% from her father. To make this easy, let’s divide by 2 for every generation.
So how much of your great-great-grandmother’s DNA are you likely to have? Probably around 1.5625%! And that may not be enough to detect Native American ethnicity.
If you can find older generations on that line to test, I recommend that. Also, get brothers, sisters and cousins tested. You never know who might have enough DNA to be detected.
Even if you find the DNA connection, you will still want to follow the paper trail. I recommend our Native American Research Guide to get you started.