A lot of people suspect that having-been part of a prestigious organisation (such a a famous university or an "elite" org in your field) gets you an unfair advantage when applying for future jobs.
There are two main avenues you could imagine for this advantage. One is basically nepotism: through the organisation you meet lots of other people who will later give you preferential access to jobs.
A second avenue is through the associtional value of the institution: that people with no specific connection to you or that organisation will see the name of Prestigious Institution on your resume and hire you because, well, you were at Prestigious Institution.
I think this is one of the situations where people are prone to massively over-generalise from their narrow, unrepresentative experiences – the relative value of nepotistic and associational benefits will depend massively both on the kind of field you're in (are there lots of other people from your particular prestigous org in it?), the subjectivity of that field (can people in it just give jobs to whoever they want without regards to merit?), and what you got up to in the organisation (did you make a lot of friends who will remember you later?)
But I do think associational value is under-rated by many people in many situations, relative to nepotistic value, and have a few observations about how that works and what you can do about it.
First: I think associational value often comes out of single-sentence descriptions of what somebody has done, and that therefore there are often relatively-easy ways to get 99% of the associational benefits of a prestigious institution at a much lower cost.
For example, in the magazine-writing world, people are often (approximately) defined by 1-3 of the most famous publications they've ever written for: "X's work has appeared in TK, TK and TK," or "this is my friend Y, she writes for [famous media brand]."
As a result, I think in many dimensions you can get the associational benefits of [some publication] writing for them exactly once, which is (presuambly, in general) easier than writing for them ten times or getting a staff job with them. And similar "hacks" exist for other prestigious orgs: if I'm right that in many situations associational value is doing a lot of the lifting, then various dumb ways of getting a nominal affiliation with a prestigious organisation can be well worth the effort.
Second: I think that one of the big avenues of associational value seems to be the "nobody got fired for buying IBM" phenomenon.
It just dawned on me with horror that there might be people reading this who have never heard of IBM, let alone an old catchphrase about them, so: basically, IBM was the big beast in tech once, and there was a popular claim that if you were in charge of buying [X] at work you should buy IBM because even if it sucked you wouldn't get the blame – it was IBM!, very reasonable choice you made, couldn't have foreseen the issues from it – whereas if you got swayed by a rival offering a better/cheaper/nicer product, if things went wrong with it you'd get the blame (and maybe fired, supposedly, who knows).
Basically, I think one of the big benefits of a prestigious affiliation is that anyone who hires you can easily justify to her boss' boss' boss why she hired you, and avoid the blame if you turn out to suck.
I'm not entirely sure how to work around this one, beyond the "try to get a mild affiliation with a prestigious institution, even if it's an incredibly silly one" hack. But it feels like if people are hiring X Grads for ass-covering reasons and not because they "believe" in X Grads, there'll be various situations where that should influence your course of action.