In Juniper's Knot the fiend says she has done many things: she has killed many people, saved children, destroyed entire towns...but none of those are the reason she is confined within the magical circle that keeps her prisoner. She never reveals why she ended up there, but only angrily dismisses the question.
In Tsukihime's side-story game, Kagetsu Tohya, Shiki occasionally refers to the haunted house his class did last year, in which he and his friend were responsible for something that resulted in their student government making an amendment that strictly prohibits tea kettle monsters, mushroom monsters, and any pot that uses the aforementioned things. The specifics are left up to the imagination.
Shiki: Yes, we did a haunted house my freshman year, too. Well, Arihiko did his part so ardently that we were forced to stop in the morning. (To himself):The Inuidake Children Kidnapping. It's an event that will forever remain in the annals of the student government.
In the Unlimited Blade Works route of Fate/stay night, it's mentioned that Assassin was once able to ward off Berserker from the Ryuudou Temple gate at one point. Since the Power Levels in the story establish that Assassin shouldn't even be able to wound Berserker, it's not exactly clear just how Assassin pulled this off - but the fact it's unexplained just makes you wonder if Assassin is really that good.
Dangan Ronpa: Monobear scoffs at the notion of explaining how he erased the students' memories of their two years at Hope's Peak Academy.
In Virtue's Last Reward when Sigma considers possible reasons for why he was forced into the Nonary Game, he mentions that he once had a one-night stand with a politician's mistress, but doesn't give further details.
During Zero Time Dilemma, Junpei remarks that seeing the base of an organ-trafficking gang had deeply traumatized him. He never explains what happened to the gang, or how he got in that situation to begin with (aside from the fact that it was somehow related to his detective work).
Likewise, it's never stated why Atlas and 2013!Atlas are in a wheelchair. From the dates, it can be deduced that this isn't related with the incident in Switzerland and considering that Atlas never attempts to undo their disability with Time Travel, it can be assumed it wasn't the result of an accident.