What has been happening is that I like to do perfectly on the Dark Brotherhood missions. When you kill the primary target (USUALLY) the "Escape before the overseer arrives!" objective occurs, and you get like 2 minutes to escape the dungeon.
Also, if you fail in your Sacrament you can abandon the quest to reset it.
What I have noticed is that if you abandon the quest after assassinating the primary target, then redo the quest and kill them again, no overseer timer appears. When you return to the Speaker, you are just given a "Failed: Escape before the overseer arrives". So, is this deliberate to "punish" restarts? If so, its the worst designed questing system in the game.