Haha, no. The more elite the institution, the more money you are expected to bring in. It is compensated somewhat by the fact that a big name institution will give you an advantage in getting grants.

There is a bit of a difference between working directly for a university and working for a research nonprofit, like Dana Farber or MD Anderson or whatever.

If you work for a university, and you don't pull in a lot of money, what will happen is that you will be expected to teach more hours to "make up for it". You are less likely to be fired. At a nonprofit, if you don't cover your own salary and expenses through grants, you will quickly become unemployed.