The earliest I can be certain anybody in my branches arrived on these shores was 1852. If I could be certain I had any Native American ancestry, even as little as 1/1024, I'd be very proud of that fact. The reason for that is I'd love knowing I had at least some ancestry with the deepest roots possible in North America. At present, we can be fairly certain humans arrived in North America by 18,000-20,000 years ago. There are sites dated older then that, but that's a conservative estimate that is generally accepted. Arrival in this hemisphere might have been earlier. Now, that's just how I would feel. I have always loved the study of American prehistory, and so it would be very cool to me to know I had even a tiny percentage of ancestry with exceedingly deep roots here. I'd also like being able to say I had ancestors here in colonial times, but I can't. Simply because I love American history, as well as its prehistory. What events in our history that my ancestors did partake in, I am proud of. I'll take what I can get.
Now, I would not identify as Native American, with such a tiny percentage, but I think it would be great just the same, I'd be delighted.