It depends on the career. For some occupations, no one cares too much about which school you graduated from. If someone is applying for a teaching job, the difference between a state school (NY state, at least; not all states really have great schools) and MIT is meaningless.
Nonetheless, that still doesn't mean the particular school can't have some influence. I'm aware of one school where the graduates are more or less black balled by many employers in that field. The sad thing is, I don't think the current students, graduates, and school have quite figured out what's going on yet. All it took was a few graduates to prove themselves incompetent in their field and no one wants to waste time with graduates from that school - at least not while there are plenty of people applying for jobs in that field.