Tolerance refers to certain monsters' behaviour in which the monsters stop being aggressive towards players after a certain amount of time (10–20 minutes or longer for the inhabitants of the Abyss) without the player leaving the area (about the size of what is shown in the minimap).
If a player leaves the area and returns, the monsters will once again become aggressive towards that particular player. This does not include logging out and back into the game but does include teleporting away (by means of random events, spells, and the like).
Jagex implemented tolerance to discourage AFK'ing, as they would then stop fighting after the tolerance begins and automatically log out. The timer stops counting if the player logs out. Therefore, logging out and in again would not immediately end the tolerance time.
Tolerance can be very annoying when killing rock crabs since the only way to attack them is for them to be aggressive, making them a dubious training exercise at Rellekka; however, at Waterbirth Island, since they are in separate spots, it is not such a problem.
Some monsters that become tolerant may become aggressive again even if the player constantly stays in the vicinity. Kurasks in the Fremennik Slayer Dungeon portray this behaviour as did the wild dogs in Brimhaven Dungeon prior to an update. Other monsters, such as bosses, do not become tolerant at all no matter how long the player remains in the vicinity. For example, the King Black Dragon found in the King Black Dragon Lair will always attack any player in his lair.
Tolerance can be beneficial to the player in some cases. Hellhounds in the Taverley Dungeon are one example. When the hellhounds become tolerant, the player can safely attack from Ranged safe spots without another hellhound intercepting them.