I'm curious too. I had one dean of admissions say that if I got accepted to another school to let them know so they could get in a "bidding war" over me. However, this was before I was accepted so I think he just meant I would get an acceptance if I got into the other school. Not sure though...
I think a more polite way would be to let them know that cost is one of your major considerations in your decision about which school to attend, and inform them of the aid offer from the other school. They will understand what you are saying and may decide to match it or do better depending on how much they want you and the overall availability of aid at their school.
Its hard to say since cost of living will always be different between cities That is a time where a "better aid package" might actually end up being the same thing. In general, you aren't going to get more grant money this way, but institutional loans might be more available.
I'd say go for it. Don't go in with much expectations either.