This was absolutely NOT the case for a student I knew a few years ago. The private institution had offered merit (but so had UConn), and when the parent called to report to the private institution that the student’s cost of attendance at UConn would be lower than the private institution might have thought, based upon rack rate at UConn, the private institution dropped their price by the merit award at UConn.
Certainly, in any negotiation, if one NICELY lets the other party know that you have a specific, better offer, and you would like them to match that offer, you’ve got a better chance than just saying, “Drop your price.”