Take a look at this report, which breaks down categories of professors by school, and you'll find that there are many schools for which contingent professors (non-tenure track) and grad students make up 100% of the teaching positions: http://www.aaup.org/sites/default/files/files/AAUPContingent...

Now, sure, they're mostly small community colleges or for-profit universities, but that is a significant chunk of the higher education system in the US.