These accounts are known as "endowments". The way I've heard it, the interest alone could pay the tuition for every student that attends those schools.
That is to say, if Harvard, Yale and Columbia (and probably dozens of other schoos) truly cared about providing an education for those who could "least afford it", they could do so without a dime being picked from the taxpayers' pockets.
I wish I knew how to ping Ron Paul to suggest he bring this point up next time this issue is addressed.
My son was recruited by Harvard. He ultimately did not get in but during the process I learned that for families with under $100K tuition is free. For those over $100K you pay progressively more of the tuition depending on income.