I should have been clearer. If the application for FA says, we can afford X and the school becomes back with Y, this is not the case of trying to squeeze the school for something you can afford.
We submitted our FA and what we can afford in good faith so the school coming back with a different amount opens the door to a discussion.
Remember that schools have a bucket of funds and they are trying to managed that allocation between the candidates applying and trying to get the threshold at which the parents can live with.
It would be much different if our request for a FA amount was X and the school gave us X and then we tried to get more.
People have such a negative connotation of the term negotiation. In fact, negotiation is defined at a dialogue between two or more people or parties intended to reach a beneficial outcome. Both parties win.