While Stuyvesant is commonly held to be the top public high school in New York City, that isn't exactly the case.
For one, there is no teacher screening process. New teachers are accepted on a simple seniority system, which means a young, enthusiastic teacher with a degree from M.I.T and an actual lesson plan based on pedagogical research will probably be rejected in favor of a demented and burnt-out veteran. I have the deepest respect for those who have braved the classroom for decades, but I am all too familiar with dementia and a disconnection from reality in those (oftentimes previously brilliant) individuals.
Then there is the problem of teachers who are just plain incompetent and are accepted through the seniority system. Mentioning specific names would probably result in swift retaliation unto me, but the general consensus amongst the students is that a number of teachers have no business teaching The Supposedly Best and Brightest.
The other two specialized high schools of the city, Bronx Science and Brooklyn Tech employ a screening process. Why not Stuyvesant? A proposed screening process was put up for voting earlier this year. It was rejected, by a landslide. The primary reason cited by most voters was "job security".
A number of other problems exist, although none as glaring as the former. The Technology and CS departments are waging war, the Student Union is meaningless, the class schedule allots equal time to Multivariable Calculus and Health class, et cetera.
Most of the above information is based on word-of-mouth and observing indirect effects of the newly adopted policies, as the administration doesn't bother to inform the students of decisions directly affecting their education and their future. Perhaps it's due to technical difficulties of doing so, although we are provided with completely meaningless handouts about non-events on a regular basis. If you happen to notice an error, feel free to correct me.