Most people who claim they are 1/16th or 1/32nd native are not at all. It is a very popular family myth. My mom told me that her mother said her great grandmother was native. I viewed that as possible since my grandmother was an orphan. I did 23andme and there is no native American. I also went through ancestry.com to build out my family tree and indeed there was no native American in there. The 60s and 70s were a period of growing acknowledgement of Native communities and I feel like that was kind of a way that people made it seem like they were at least nominally supportive.
Or it was just one of the batty things about my grandmother. Orphanages back then were simply work houses. You did school for a little bit and then went to work manufacturing. The discipline must have been pretty strict considering one of the teachers beat her so bad that she lost an eye.