It's actually quite common for white Americans to claim Native American ancestry, especially Cherokee, and especially since the late 1960's.
This article explains the background, and this is the main reason.
I don't think most folks would admit that, and many carry on the myth in good faith. I'd say probably half the white people I grew up with claimed native ancestry, somewhat in good faith. Maybe it was a tale from Grandpa, embellished as it was passed down or a more palatable explanation for "non-white" features than believing your ancestors might have been African, Asian, Arabic or Jewish, for example. To be fair, if you genuinely had native ancestors and "passed" as white, you wouldn't have shouted about it until maybe the 1960's, when Native American art, music, dress, culture, etc. was appropriated by the mainstream - and when Native Americans were no longer seen as a threat to white "civilisation."
A good friend from High School had one of those Ancestry DNA tests done a while back and it showed he was just Irish, Scottish and English. His sister was furious because she'd built a "New Age" type business on the back of claiming Native American ancestry.