Does each Tribal Nation have its own distinct criteria for determining who is eligible to be recognized as a member of the Nation? And does such criteria sometimes vary from one tribe to the next?

I've always wondered what someone would do if their immediate parents and grandparents all happened to belong to different tribes? Is it theoretically possible that someone could have an ancestral mix of so many different Tribal Nations in their bloodline that none of those Nations are willing to grant the person membership? And, if that happened to be the case, how would they be expected to self-identify on Census forms?