I think Clover has it right. You can't know unless you try.
I had my undergrad in Business, from an obscure but strong regional university and fell in love with Anthro to broaden my background towards International Business. I applied to Oxford for an MPhil in Social Anthropology and by some miracle was accepted (meanwhile I was rejected from three other schools with much lower profiles).
To summarize: You never know...give it a try!