I regret choosing an elite university (Oxford) and its ultra-generic Engineering course, rather than a more specific course in say, electronics or even maths but at a less prestigious institution. I came out with a paper-thin knowledge of a dozen engineering subdisciplines, rather than truly knowing any one of them. Many of my peers became accountants, many others quickly became 'project managers' - spreadsheet jockeys. I cannot think of any who actually became engineers. Like others, this frustration has led me into a life of continuous learning.
Funny, I regret going to a plain state-funded university and wish I would have taken college applications more seriously to take a shot at a more prestigious school. I think a lot of doors would have opened earlier for me had I done so. Successful people get there because of their network and connections not because of their knowledge and skills.
If you're a regular on this site, prestigious schools are mostly bullshit. They're not good places to go for an engineering degree, except MIT and Caltech maybe (and even then they're really not the greatest places to go for a BS).
If you're a politician or trying to become a corporate high-up, then yes, the network and connections you make at the prestigious schools are extremely useful. But someone who's a CEO or VP of a hedge fund or whatever is not going to be a regular on this site.
If you want a career where you're valued for your knowledge and skills, good state schools are the place to go. If you want a career where you're valued for your network and connections, ivy league schools are the place to go.
In a similar vain, I don't regret "choosing" a non-elite uni (Oxford Brookes) which let me miss lectures and tutorials for 3 years, tinker with my own code all day, showing up only for exams and the final year project ("thesis"-wannabe-thingy) for a proper B.Sc. graduate degree. I did sometimes go to Ox open lectures and such, loved the overall high-brow intellectual vibe in town back then (2002-2005). But the workloads and schedules of my acquaintances there, oh my!