The first time that sleep overtakes the narrator is during his effort to circumnavigate the walls of his dungeon and thus find out how large it is. When he wakes up, he finds a loaf and a jug of water beside him, which indicates that he has been observed or inspected as he slept, and shows us how carefully his keepers are watching him.
The second time he sleeps is after he discovers by accident that there is a hidden pit in the center of his dungeon, but escapes falling into it when he accidentally trips before reaching its edge. Again, when he wakes up, he finds a loaf and water beside him. However, this time, the food and drink appear to be drugged, so that his keepers can manipulate him freely while he is unconscious. They bind him to a "low framework of wood" which is intended to hold him in a fixed position as the pendulum slowly descends from the ceiling and cuts him in half.
The intervals of sleep thus serve as punctuation to the narrator's ordeal, allowing the narrator's keepers to first challenge him with the pit, and then with the pendulum.