Most schools say they don't like to "negotiate." But if you can phrase it in a way that doesn't make them feel like they are in a used car lot, you might succeed. Also, schools are more likely to bid up aid for students they feel are especially attractive (i.e. might raise their average test scores, or help a sports team win a championship or something.) But if your student doesn't bring anything really special to the school, they are much less likely to throw any extra money, since they will want to save that for students they really want. One way to phrase it might be to simply ask if a more generous school perhaps understood your family's finances a little better?
Here's an article I wrote laying out some strategies that have worked for other parents:
A Few Simple Steps and Some Tough Questions Can Help Families Tackle the Cost Of College - US News and World Report
And if you go to
Financial Aid Letter - Letter Decoder and click on "action plans" you can find some step-by-step instructions. Good luck!