I am a current graduate student at NYU and I have lived in Brooklyn for 2 years. I currently occupy a small one bedroom in an outskirts-neighborhood, and my $1200 rent is considered a steal! I would budget at least $1500/month especially when you factor in the brokers fee which is very hard to avoid paying. One bedrooms in "nice" areas like Park Slope and Brooklyn Heights will run you over $2000/month :-(
My utilities (electric, hot water, internet) average $150/month and I am lucky to have my heat included - which is not always the case.
Luckily, I split the expenses with my significant other - otherwise it would be very tough. I only know one one graduate student who lives in NYC without a roommate, even with NYU's MacCracken. But that doesn't mean it's impossible! And I know that there are many European cities where the cost of living is higher.
Best wishes to everyone!