It's too bad that I couldn't have gotten a DNA sample from my grandma, who, if my great-grandfather was correct about his mother being Cherokee, would have been 1/4.
This might be a little off-topic, but it sounds like you're interested in this area of research, so I thought I'd share a few things.
First off, I majored in anthropology and minored in history and sociology in college, and took graduate courses in archaeology and museum studies, so I've studied this from a bit of a different angle than most people. I learned that ALL the earliest Native American remains found (at least in North America at that time; Class of '03) have Caucasoid features. One of them was said to greatly resemble the Japanese Ainu people, or the actor Patrick Stewart (this was Kennewick Man, found on the bank of the Columbia River, originally thought to be the skeleton of an early settler before they found a spearpoint imbedded in his pelvis and did a Carbon 14 test...don't quote me on exact placement of the spearpoint; it might have been in his thigh, but I'm not looking it up right now). Another example is Spirit Cave Man. So I am not surprised by this information about a common ancestor.
I thought you might be interested in books by Gavin Menzies. He hypothesized that, in 1421, China sent a fleet of ships around the world to map it and establish trade relations. They reached the Americas, and pretty much everywhere else, too. There is archaeological and linguistic evidence for this, as well as genetic evidence. He has written four books, and, as time goes on, he keeps gathering more data and pushing back the date for when these voyages took place. I am currently reading his book on Minoan civilization, and I think he is going to say that Minoans also were all over the world (including the Americas, mining copper in the Great Lakes region). I have not had the opportunity to vet his sources yet, but if he isn't completely fabricating and stretching everything he says, he seems to be on to something. This is relevant to genetic genealogy.
Coming around to my point,I have read that people with ancestors on the Cherokee rolls sometimes show up as not being Native American at all, but rather, that they have Middle Eastern or Jewish ancestry that they did not expect. There is something quite off, because we just don't know enough about ancient human migrations to attribute ancestry to the correct population across the board. There are going to be weird little quirks and inconsistencies, probably for a long time.
I have read some very fringe (IMO) ideas about Cherokee not being Native American at all, but rather, being descended from Eastern Europeans (i.e. Albanians or Georgians) who fled from the expanding Ottoman Empire. Melungeons claim to be descended from shipwrecked sailors, probably from Portugal, but I have to wonder if these could be the shipwrecked Chinese sailors that Menzies wrote about. This all ties into who the Cherokee were, genetically.
It's clear that there was Mesoamerican DNA in this region (Georgia, Tennessee, etc.), and it seems hypothetically possible that this might be the entire Native American contribution to an otherwise pre-contact (with British immigrants) population of Eastern European refugees, shipwrecked Chinese sailors, and Native Americans who were already living there at the time that these other groups arrived, intermarrying for a couple centuries before the Western European colonists arrived.
It all seems pretty fringe to me, but DNA doesn't lie. Time will tell who is right and who is speculating inaccurately.
I would love to know more about the migration patterns of ancient humans. Back when I was in college, we could really only follow the linguistic evidence and the artifacts. Now, we can follow the DNA, too, and I think we're going to find out some very fascinating and unexpected things.