"A dear friend and excellent negotiator told me that when he gets any kind of short-term exploding offer, the first thing he does"
As someone who considers himself an excellent negotiator (and gets paid to do negotiating) [1] I'd really caution someone from employing a technique that someone else uses that you don't have experience with. Negotiating is all nuance it's not engineering or an exact science. Go try and read a book on it if you want. (I can always tell someone who is fresh from reading a book.)
Actually exactly for this reasons:
"I don't want to be responsible for anyone losing a deal"
Of course, it's easy when someone else is taking all the risk.
"when someone offers you an exploding offer, it's because they really, really want you to take it."
Could be. But there are cases where this is not true. How lucky do you feel in interpreting all the signals based on a limited time as a negotiator?
And getting back to what Sam has said:
"Sometimes they say they have a fixed amount of desk space, but in practice, if a good company wants to join late, they always make room."
Things are never always clear cut. Very possible your company is borderline and you don't fall into a "must have" category. By even YC's own thoughts there are many more qualified companies than they can accept. Not everyone is a superstar that can write their own terms.
"Any deadline claim has to be concrete and believable."
How so? You ask them? You investigate? You poke around? General advice the specifics are much harder.
In any case the last thing you want to do (according to at least my method which might differ from another negotiators method) when you are negotiating is tip your hand. And if someone is bluffing the last thing you want to do is let them know or that you think that they are bluffing. As such saying anything that sounds like "prove it" is certainly a way to do that.
In negotiating often people claim something that isn't true and bluff. It's part of the game. The idea is not to make them prove it. The idea is to figure it out on your own and use it to your advantage.
[1] It's a game.