I agree with most answers here, but wanted to add another point, purely from the standpoint of a career starter when your education is weighed more heavily: just because a school isn't among the "big names" doesn't mean it doesn't have a reputation.
I would mostly advise you to beware of attending a Uni that has a reputation of not being credible overall, or one that does not offer a credible program, given your major area of study.
For example, majoring in computer science at a school well-known for it's art and theater programs is not likely to help you land a programming job, and it might raise a small suspicion in the backs of people's minds. Likewise, a school can be unremarkable overall, but it may still offer a well respected program for your major. In this case the back of the mind will feel more satisfied that your education level is acceptable or even superior, depending on the program's reputation.
Meanwhile, attending a "party school" is not going to appear as credible in a more general way, so unless that school is known to have an outstanding program in your major, attending might not do you any favors.