At Brown, students are actually on the adcom, so they do carry some weight. However, I doubt a student's opinion will count as much as the assistant dean. Even unofficially, power plays a part.
The student's impression is not as important. However, if the student feels strongly that you are not a good fit, then I'm almost certain you won't get in. There are too many good applicants. They are not going to take a gamble on someone who had a bad interview.