I'm not sure how things are for MPA programs specifically, but I think it depends on the department and the school. I had a friend in engineering get into a masters program in July last year.
I'm applying for statistics masters programs and applied to a few rolling admissions/late deadline programs:
Brown (rolling, applied in March), Columbia (rolling, final deadline June 2), Georgia State (July 1), and Troy University (rolling, but I applied in Feb and haven't heard back).
I doubt our chances of getting accepted are that much lower, especially since it wouldn't be fair for programs with later deadlines to fill up their class early on and still accept applications from everyone else. I also don't think funding and assistantships are out of the question since many programs decide on funding over the summer/beginning of fall, but once again, it depends on the department/school.
Best of luck to you!