I always find these kinds of topics funny. I encourage a different set of criteria for what one claims or doesn't claim: do you consider yourself to be a Native American? Have you had interactions with the culture or have you experienced discrimination because of your race? If either of those are "yes," then I think it is more than reasonable to mark it on an application.
As far as tribal enrollment is concerned, it is a crock of *&^% how it works and is not really indicative of what a person is or is not. Trust me, I have personal experience with this. Tribal enrollment can be absolutely dependent upon treaties signed over a hundred years ago, and many tribes have set up exclusionary policies to keep a stranglehold on federal money and casino revenues...it is entirely possible to be 100% Native American but of various tribal blood quanta and be unenrollable in any tribe. Would that mean a person should not or could not claim Native American simply because the federal government's arbitrary tribal recognition system? Hell no.
Put down whatever you feel you are and, if asked about it, explain clearly how and why you are, regardless of enrollment or lack thereof, a Native American. Remember people, we are not pieces of paper, we are flesh and blood.