While ultimately I think the student has the greatest impact on how much they know, I would have to argue that your institution will have a profound effect on how much you know.
First, it impacts the opportunities you have to be in learning situations. Second, it determines how the material is presented to you, and third it determines who will present the material.
Under normal circumstances, more opportunities to learn and participate is better than less opportunities to learn and participate.
Having material presented in a form that is logical and consistent with your style of learning is better than having information randomly thrown at you in a manner that makes no sence to you.
Having instructors that are approachable, concerned, and competent is far better than having snobby, preoccupied, and/or incompetent instructors.
Laslty, your school is generally going to be the definative on WHO you know. Sometimes that goes a long way (sometimes).
The problem is as pre-meds we tend not to know what to look for in a medical school. So, we let US News & World Report, our peers, and "not-so-common sense" dictate what is important.