It depends on the degree. For STEM, not so much. For law, business, humanities, absolutely.
Of course, you could always say that this is a function of inherent talent and drive more than institution, which is possible, but even then it's just a vital social signal and leaves out the 99% who could never stand a chance of getting into elite institutions like me.
It very much depends on the degree. I don't see any Nuclear Engineering luminaries coming from Harvard, and no more coming from Stanford.
- An NC State Nuclear Engineering alumn
At Stanford, you get a good technical education, but you also get contacts that will become important players in tech. At Harvard, you get contacts that will be important players in government, business, and media. At Random Public University, you get contacts who become players at the state level.
I graduated from NC State in 2000 and I've had a really awesome, rewarding career filled with interesting problems and fun. As an engineering manager I've had many, many colleagues who have advanced degrees from fancy schools that I never could have gotten into (or afforded). I use stuff that I learned at NCSU in my job every day, and am extremely thankful to have gotten such a high quality education at such a bargain basement price (in-state tuition was very cheap, I used to put it on my credit card). I've never felt like my degree or lack of a fancy school has hindered me.
As a hiring manager, unless the person is coming directly out of school, I don't even look at their education. There's a great many people that I've hired where I had no idea if they had a degree at all, much less CS.
The order in my mind is:
1) STEM elite 2) Scholarship elite 3) STEM local 4) Any other program local 5) Trade 6) Any other program elite
This order does not apply to wealthy people. Those who come from wealth can apply to an elite college and get an elite education and work for their friends without stressing out about school debt.