There's evidence that New York's famed magnet schools--Stuyvesant, Bronx Science, etc--do not improve outcomes for the students who attend them. If you look at the marginal kids who just barely made the cutoff on the entrance exams, and compare them to the marginal kids who just barely missed getting in, there's little difference.
Consider it your daily reminder that when it comes to education, good outcomes are not the same as great teaching. The most reliable way to amass impressive alumni is to screen for impressive freshman. But at the policy level it's more important to identify institutions that are unusually good at helping people learn, not institutions that are unusually good at screening.
This doesn't mean that schools don't matter at all. For starters, there are almost certainly threshhold effects--Stuyvesant is about as good as the sort of high school that the kind of kids who can get into Stuyvesant usually attend. Most of those kids are probably not coming from the South Bronx.