I know this thread was posted as bait, but there are people on this board who might take the Ivy League meme seriously, so I'll offer a different perspective.
I went to a tiny regional university in a flyover state no one cares about after they offered me a full ride for my good performance in high school. The class sizes were on par with a good high school, so I was on first name terms with most of the professors in my department.
The tiny student body and ease with which I could get to know my professors made it easy to get research internships, so I graduated with a STEM degree and three full years of research experience under my belt with no student debt to pay off whatsoever, and I had multiple professors willing to be professional references for me as I looked for jobs afterward.
Small universities can be fantastic if you're able and willing to make positive impressions on the people around you, since it's much easier to stand out in a classroom of 25 than in a lecture hall of 250.
Make sure the school has a good program for your desired degree though, otherwise you'll likely get a subpar education just because your environment won't have been conducive to learning what you need to learn.
Of course, my experience isn't indicative of what everyone's should be, especially since I didn't pursue grad school. If you want to go to grad/medical school then prestige can be very important. My best friend in undergrad had a 4.0 GPA, just as much research experience as me, belonged to three different honors societies (one more than I did), and worked at a nursing home in her spare time get into an Ivy League med school where a lot of her colleagues got in by flashing their Ivy League undergrad degrees and/or their parents' money.
