Do colleges lower their fees to match tuition assistance?

I haven't been able to find this answer anywhere. I want to use tuition assistance but most colleges are more expensive than the credit cap of $250 ish that tuition assistance covers.

Is the yellow ribbon program only for the GI Bill or does it often have a similar effect lowering tuition for tuition assistance? If not is there any similar program or way to tell if I don't pay out of pocket?