Well, there's a couple times when such things happened for me:
In SoulSilver, the only one with a type advantage against Will was my Arbok. Knowing it'd likely be one-shotted by Psychic, I gave it whatever berry weakens super-effective Psychic moves. His Xatu uses Psychic, and the berry activates. Arbok is down to 1 HP, and I was overly impressed by it, proceeding to one-shot Xatu with Crunch.
In X, I was capturing Xerneas. It's down to the red, and I try an Ultra Ball. It fails. I try a Great Ball. It works. My lesson learned here: Always carry some Great Balls and Poke Balls around, even if you have no intention of using them.
Mewtwo, who I expected to be extremely tough to capture, was less so. It was easy to weaken, and caught it within less than 10 turns. I was happy about that.
In SoulSilver again, I was preparing for Mewtwo, then remembered its only attack move was Psycho Cut. I bring a low-level Poochyena to help weaken it bit by bit, until I lowered it down to near no HP. I try all sorts of Poke Balls to capture it (Ultra, Great, Timer), then on the first Fast Ball, I capture it. I was both happy and upset, because, for one, my Fast Balls never work out. Also, it messes me up to remember such things happening, and RNG sometimes sucks.