I am a 1/5 Native American and at some point spent time on our reservation, but because I no longer align myself with that culture; i left it off of the "check-box". I touched on it briefly in my diversity statement, but I don't think that if you have never been affiliated with the culture you should claim it.
I have to disagree. Do you really think that being 1/5 Native American has affected you in no way? When people ask you why you look the way you do what do you say? 'Idk I'm tan.'???
There are definitely varying degrees of self association and I know some girls who are less than 1/16th but really associate themselves and their families with their tribe and I think they're totally legitimate in claiming to be native american. For instance there is a girl at my school who participates in all of the NA club activities etc. and she and her family are very very involved in their history but the girl AND her mother both look like they just stepped off the boat from Sweden.
I think it's unlikely though that having a grandparent being native american hasn't affected you at all. I think you should check the box if you want and not feel bad about it.