Not in my case (Caltech). It was clear when I got an industry job that a BS from Caltech was at about the same level as an MS from other institutions. But if the institution is large enough, one can get an elite education by careful selection of the classes and professors, and doing the maximum in those classes rather than just getting by.
For example, take honors calculus, not weeder calculus. Don't avoid the math heavy classes. Don't waste your time with easy-A classes. If you don't find yourself studying 6 hours a day, you're not getting an elite education.