It depends. Some schools have programs that are specifically designed for students that have just finished their B.A. (Chicago is one example). Other schools are different, and are designed so that it really doesn't matter one way or the other. Chicago as a whole accommodates M.A. students, but from what I've gathered the English program is primarily designed for students who have just finished their undergraduate degree(s). I've never heard of a school rejecting a student, though, because they already have an M.A. In terms of already having a B.A., many schools might possibly reject you (if, unlike Chicago, their program isn't specifically designed for finished undergraduates); NYU, I recall, says that they are more than willing to consider a student if they just have a B.A., and UBC says that on certain occasions some students can go straight from the B.A. to the PhD.
But if the school isn't like Chicago in the scenario mentioned, it is always a possibility that there are students with M.A. degrees who have a bit more research experience, publications, teaching experience, etc.