I think it's plausible to be told something about your heritage and not question it, especially if it's something spanning multiple generations where the paperwork validating the claim is either nonexistent or otherwise hard to reach. At varying points I've been told by relatives that I'm part Scottish, and told by others that I'm not, and I don't really have a straightforward way of validating this aside from engaging in some intensive research.
I can reasonably extend the benefit of the doubt to Elizabeth Warren that she didn't know if she was actually some fraction Native American or not until the DNA test.