If you choose NYC, you don't have to rent an apartment in Manhattan. You can rent an apartment in Brooklyn, Queens, The Bronx or New Jersey(not really sure about this one). I believe you will find a good place for much less than 3500-4000 (as it says in the link provided by Tsotne).
Depends on the size of the apartment, but I believe you can get something decent for a student for $1500- $2000. By the way, I think its possible to find an apartment in Manhattan for less than 3500-4000 but I'm not really sure about this one :\