Well, it may be a little different in the UK due to our many-tiered higher education system (Oxbridge, redbricks, ex-polytechnics and a moderate spectrum in between), but I'll chip in my two copper pence either way.
Actually, this is a question I've thought about a little too much, I'd wager, because, as a prospective undergraduate, I actually turned down an offer to read mathematics at Cambridge and decided to study at the University of Bristol instead. The latter is in itself a well-regarded institute, no doubt, but it cannot be said to have anything near the Oxbridge 'name-weight' (for want of a better term).
Do I regret my choice? In some ways yes, in some ways no. The decision was very much informed by my particularly working class background - I did not believe I would have 'fit in' to the predominantly upper-middle-class culture of either the university or the city that surrounds it. In hindsight, my opinion hasn't changed at all. Bristol suffers many of the same problems as a university, but as a city it is as colourful as they come.
I believe that forgoing the 'top tier' university worked such wonders for my social development (and, hence, my subsequent happiness) that I cannot bring myself to regret it wholesale. However, I can't help but feel there's an undeniable magic that happens when a highly selective and competitive institution brings a bunch of smart, young people together to explore their interests.
Being a part of that could, as the "Disadvantages of an Elite Education" article suggests, have ended up insulating me from the 'regular world', but there is the odd occasion where I cant help but wonder what could have been if I'd chosen to experience it. Maybe that's what it takes, you know? I try not to dwell on it for long, though - I've left that to my folks!
I've not planned to become a high-ranking politician or appear on BBC Radio 4 (yet), so a non-Oxbridge education hasn't hindered me much so far... Either way, though, I'm still (relatively) young and it will be a good few years before I can evaluate these decisions with any great degree of accuracy.
We'll see, I guess.