For what it's worth, I was offered a particular scholarship only after declining the school. The dean of admissions called me up the next day and asked me to reconsider.
They can play their hand of cards anytime they want. If they can get away with you accepting their invitation with the least amount of money in your pocket, that's money in theirs.
I responded by telling them that the school I already accepted was still less expensive after all was said and done, so they asked to get back to me with a better scholarship.
The moment they're aware of competition, some schools will feel a lot more generous.
I ultimately stuck with my original choice for multiple reasons. One reason was that I felt disrespected for only being offered the scholarship after declining the school to begin with. So at the same time, try not to disrespect the school offering you the scholarship, but don't waste the opportunity to bargain while the money's still hot.