The trapped in Pokémon Mansion oversight occurs in Pokémon Red, Blue, Yellow.
The problem is notable for affecting Twitch Plays Pokémon during the initial playthrough of Pokémon Red.
On the third floor of the Pokémon Mansion, there is a Burglar that walks left and right. It is possible to position him one tile above a 1x2 gap, effectively trapping you if you initiate the battle with him by talking to him one tile south of him from that position. Once he is defeated, he stops moving, meaning it is impossible to leave the Pokémon Mansion without blacking out or using an Escape Rope/Dig. Saving and resetting still doesn't let him move.
This problem still exists in Pokémon Yellow, without a solution for those who don't have Dig or an Escape Rope, except blacking out, though that could take time.