late edit: DISCLAIMER: The pictured map is not actually a representation of the territories before colonisation. It's a hypothetical map of what countries there might have been had the continent not been colonised, thus all the names and borders are fictional and have never existed.
For good actual maps, check out native-land.ca.
That was me. Grew up thinking that one of my great grandparents was full blooded Cherokee. Took a DNA test only to find out that my DNA only had like 1 or 2 percent Native American DNA.
To be fair, those kinds of numbers are consistent with a Native ancestor maybe 5 or 6 generations in the past. Blood would be diluted by half each generation.
If you assume 20 years/generation, then that's like 100+ years before you were born. Maybe some time in the 1800s.
Of course that's an idealized scenario, real life is more messy and uncertain. For example if 2 people that were both half native had children, then the children would also genetically appear to be half native.
Also you never get perfect 50% splitting of genes over many generations because of how genetic recombination works when egg & sperm cells reproduce. The result is that there's a chance you contain 0% DNA from a distant ancestor. It's like a 5% chance you have no DNA from a 5x great grandparent. Keep in mind, for an idealized family tree you'll have more than 100 ancestors at the 5x great grandparent level.