In the US, each $1.00 added to federal funding results in colleges increasing costs by $0.67 - so yes, what you suggest might happen.

Bernie is back to his socialist BS again, wanting free college, health care, and a bunch of other things, all for the low, low cost of more than $33 TRILLION over 10 years.  That doubles our spending,  and would require us to double taxes, all for nothing because it will just drive costs up faster.