On this topic,
I got a very solid piece of advice from a faculty member. Play your cards with the schools that are in the mix, but only approach schools you are willing to "walk away from" if they don't offer you any more aid. They have to be "last resorts", so to speak. It would be hard to say, "I can't come unless you fund me ... " and then, upon getting no more funding, still attend, right?
Hope that helps. It'll be my strategy. But, I should say, I've talked increased money availability with a director from a school at the top of my list, but only after frequent personal emails...
Hope that helps.