A student most have something for the school to work with. If the student is in the lower range of their acceptance, they may only be able to offer what they can offer. If a student has good grades and test scores plus maybe some extra curricular's (not athletics) then the school may have something to work with especially if the student has other suitors.
Also do not take the first offer as their best offer. We went back and forth between several schools and my ask was to get me to my EFC. No school could, Except one. But when push came to shove Son did not want to attend that school and for a few thousand a year, I preferred that he went to his first choice.
Do not fall for we will match your best offer. When we went through that we found out it was for all schools except one. You guessed it the one that offered to meet our EFC. These coaches know what the other schools cost and know that they will be in the ball park. Another schools offer may not be the matching schools best offer. You want each schools best offer. I heard a lot of no's. But many times when going back no's would turn to yes'. I would not accept no until I heard it twice.
A coach may or may not have pull in financial aid. If they do it is usually not much.