I've been in this position for undergrad. I got about $400,000 in scholarships, both from W&M and from outside scholarships. I only get back a check each semester for the difference between total expenses (tuition, room and board, , etc.) and the total Cost of Attendance (COA). In general, you cannot get back more than the COA for an institution. (But for me, all the extra money went to my mother back home, so I still have $3.50 in my bank account.)
So what happened to the rest of my money, you ask? It just gets dissolved in the system. No joke.
Sometimes, though, the individual scholarships are made out to you (not the institution), in which case you can cash away.