I know that it can be done. I've heard success stories, but I haven't tried it myself yet. I might be soon. I figure that it's worth a shot, right? The worst they can do is tell you "no."
I leveraged my Vandy scholarship to get my Texas scholarship, which I leveraged to get my Cornell scholarship. I also think I could have used those three to get more money from WashU and W&L for sure.Basically I had to fax UT and Cornell my scholarship letters, and they responded accordingly.
Yes, that's exactly what you should do. Just make it clear that you really like their school, but that financial considerations are getting in the way.