What sort of fraught moments have you experienced? What were your conflicts and what were the resolutions?
I can think of a couple, one recent and minor, another from years ago and much more significant.
The recent one was not in D&D, but in a game of Stonetop. It's a PbtA game about the champions of a quasi-iron age village. While there are certainly elements of challenge, the focus of play is more about discovering what happens to the characters and the village.
The issue came when I introduced a complication on a partial success to a player who can at times be very win-oriented. He's a huge video gamer, and that can bleed over into his RPGs at times. He was trying to get help from one group of people against another, and for the complication due to the partial success, I introduced a level of uncertainty about the new allies. That although they will help now, they may cause issues in the future.
What happened next was that he locked up and couldn't make a decision to accept the help or not. The other players were all watching, and kind of indirectly urging him to proceed. But he couldn't accept anything but a perfect resolution... unless he got assistance without any drawbacks, his brain couldn't proceed. The other players got frustrated, and then he got frustrated in response. We resolved it by calling it a night and agreeing to talk about it later on. A couple days later, he and I met for lunch, and we discussed it. We talked about the purpose of play and expectations and so on. And things have been fine since then.
The other time was far worse. It was about 12 years ago. My group was playing Pathfinder at the time. I was the primary GM and I was growing increasingly frustrated with the system and the process of running it. I had a player who was very much focused on character builds. He's spend hours in between sessions working on builds and trying to come up with really combat-effective characters. Then he'd bring them to play, see them in action for a session or two, and then his interest would totally die. And he'd ask to make a new character. After this happening several times, he came to the game with a character that was one of the most absurd things I'd ever seen. It was a version of the Summoner class from Pathfinder... except the Summoner would summon the Eidolon, and it would surround the summoner like power armor. Not bad in and of itself... but he could also somehow duplicate this thing so that he had another? Something like that.
Anyway, it was a really broken combo of things and his character was just like super effective. In a tough fight against some really tough enemies, the Eidolon actually was defeated (or banished? I forget, but it was out of play). The character then proceeded to break out his bow and was as effective an archer as our ranger character. So then after several rounds of like 6 shots per round or something, I eventually asked him how many arrows he had.
He totally flipped out. Got mad at me because "we never track ammo" and so on, and how I was just trying to find a way to "make him lose". He stormed out.
Now, I'm not going to say it was entirely his fault... I definitely bear some of the blame. I was not trying to make him lose so much as trying to find ANY weak spot to his character. Like some kind of drawback. The fact that I got to "how many arrows do you have" shows that I was scrambling for any kind of shortcoming. So, he was right in a way... I probably was focused on that stuff more than I should have been. On his side, I think he was approaching the game from a way that was totally over the top and extreme.
We smoothed things over and remain friends, and he still games with us upon occasion (he's since moved, so it's rare). I also have never run or played Pathfinder again. It was the last straw for that game for me, and probably a sign I should have punched out sooner. I don't like its focus on character build and the way that works and how the system allows for truly absurd combinations.