It's either a mixup of game speed, or people just being idiots on the internet pretending they're amazing by just making stuff up. This happens constantly on the forums here, I mean we have people claiming to clear whole maps by turn 150 or winning by science on turn 200, and it should be apparent that these people are full of it.
I mostly play MP so I'm used to quick speed, and I can tell you yes you can get the national college by turn 60.. sometimes turn 50. But that is only when you DO NOT build a settler first, and instead build growth buildings in your capital (granary, watermill) and go straight to philosophy. I don't see how it could be done with several settlers in first though, although turn 70~80 with 2 or 3 cities can for sure be done, given the right conditions. This usually involves settling your 2nd and/or 3rd city on a hill with a plains tile you can work. With liberty, a city like that pumps out a library in 10~12 turns. So yeah, if you rush the settlers out and put them in spots like that, you can usually get the national college queued up in your cap around turn 55~60 and let it build. But no, any claims to have 4 city national colleges up on turn 60 are pretty much guaranteed to be epic lies. And this is at quick speed I'm referring to... if they're talking about standard there's just no possible way.