Partially true but there are many other reasons why. Personally, I always thought it was administrative bloat which in turn put higher costs on the students. A 2015 study[0] shows that the real culprit could be as simple as less state support per research institution.[1]
>more than half of core educational expenses at “public” 4-year universities are now funded through tuition, a private source of capital, they have effectively become subsidized private institutions.
[0] https://www.demos.org/publication/pulling-higher-ed-ladder-m...
[1] https://www.demos.org/sites/default/files/imce/Pulling%20Up%...