I've played it out a few different ways over the years. We've talked about that a bit up thread, but here is a case I have in mind.
The PCs are taken captive in what would otherwise have been a TPK (as often advised in official published material.) The NPCs want to pry crucial information from them, which the whole party have promised one another to withhold. We're not interested in torture-porn so we have no interest in narrating the process in any detail. Each player is now put in an awkward position? What decides if their character spills or not? In the end, the players found it to be a relief that I let the dice decide. It made perfect sense to them that they might have been able to say nothing, but there was a chance they would break. It avoided any player having to intentionally do something that might have felt at odds with their commitment to the party.
This certainly isn't the only way to run the situation, yet unless the situation is something a group are anxious to narrate, it is a decent way to do it. I wouldn't mind a DM saying - yes C you hold out as long as you can and - make a roll - you give up the location of the prince on the 3rd day. If a DM asked me if I caved, I think I'd prefer to roll, in most cases.
The bolt bits are essential here: if the players are buying into the dice, then there's no debate. In fact, if we're talking about situations where the players and the DM agree, this entire conversation is moot*. I have solely been considering situations where there is disagreement.
Just for fun, here's how I imagine this unfolding:
I've just stepped in
deus ex machina and saved the characters from dying. Therefore, their souls (and character sheets) are by rights mine. I would have no problem making the information the price they pay for this. "Without going into gory detail, the king will torture the information out of you. If you refuse to talk, you'll need a new character. What do you do?" I don't need to roll dice...that's going to be intimidating (unless the player hates their character and wants a new one). Now, if the player thinks, "You know, I honestly don't know if my character could hold out" and wants to roll dice, fine. But that's just using RNG to assist in making a decision that is theirs to make however they want. Or if they want to be the hero, I'm fine with that, too.
One interesting variation is they decide to lie. That in itself might call for a dice roll (Cha? Con?), but probably another dice roll (Int? Wis?) to see if they all tell the same lie. And if they fail the second check....wow, not sure. It gets complicated.
Or another variation...assuming everybody buys into it, because this is the sort of thing that could wreck real life friendships in the wrong group...is to set it up Prisoner's Dilemma style.