I looked up what Stony Brook's cost of living allowance is and it is $14,228.
http://www.stonybrook.edu/finaid/cost/graduate.shtml
Concerning the $20,500 in stafford loans, this amount is the total amount that the government will lend you at 6.8% interest. $8,500 is subsidized, which means the government pays the interest while your in school; and $12,000 is unsubsidized, which means either you pay the interest while in school or let it accumulate and pay it when you graduate.
The tuition looks like it is about $13,000 a year (which, by the way, is very reasonable) if your a NY resident. so your total financial aid package should come out to ~$27,000 / yr if you are a resident, more if you are not.
Since the $20,000 in stafford loans are not enough to cover that, you can take out an additional loan, like a grad plus loan or others, at a slightly higher interest rate to cover the rest. These loans, of course, are not subsidized but you can let the interest acculmulate while in school and defer the loans until you have graduated if needed.
Hope that helps, But follow up with a financial aid advisor as some things could be different at all schools, and I only briefly looked at the website.