The military is also a non-profit institution, yet it's the most expensive thing we pay for in the US. Just because the institution itself is non-profit doesn't mean there aren't tons of entities making a killing off of growing the beast.
A business need not be corporate-profitable to pay its management lavishly. The university may not have shareholders or a dividend but that just means all the money it makes has to go somewhere, and that is often inflated salaries for the highest echelons of the school.
What about the fact that most of the best-known schools have basically become massive hedge funds with nices educational institutions attached to them to keep the non-profit scam in play?