In the past few years, schools -- especially the selective ones -- have become more flexible about their initial offer. We heard one FAO urge a group of students to get in touch with her if they were considering another school. "Perhaps we overlooked something in your circumstances," she said. What she meant of course was, "Perhaps we want you so much we will be willing to increase our offer."
Before you contact the school, gather all the supporting ammunition you can in front of you. If you've received a better offer from a comparable school, have it in front of you when you call and be prepared to send a copy of the rival award letter to the school with whom you are negotiating. They will probably ask to see it, which brings up another point: Don't lie.
If you feel that the school has not understood your financial circumstances, be ready to explain clearly what those special circumstances are. Any documentation you can supply will bolster your claim. If your circumstances have changed since you filled out your need analysis form (for example, you have recently separated, divorced, been widowed, or lost your job), you should be frank and let the FAOs know. They will almost certainly make changes in your aid package.