Symbolism, Imagery, Allegory
After Bud's mother dies, he has a tough time sleeping. Like Bugs says, "… everybody knows how you like to sleep with that knife open so I figured it's best to grab holt of you so's you wouldn't wake up slicing nobody" (8.7). Bud has some pretty rotten things happen to him while he sleeps in Bud, Not Buddy (like that pencil up his nose) so we don't blame him for keeping one eye open. But his sleep pattern changes once he meets Lefty Lewis.
After Lefty gives Bud food and a ride, Bud covers himself with Lefty's jacket and passes out—without his knife open. We find out the next day that he slept through a run to the hospital to drop off blood, and he didn't even wake up when he was carried into Mrs. Sleet's house and stripped of his clothes. He sleeps in a bit, too, which is weird for him. He even sleeps until noon the first night he stays at Grand Calloway Station, which is the first time he's ever done that (16.27).
Why does this happen? The way Bud sleeps tells us how he is feeling. When he sleeps really deeply and for a long time, we can tell he is feeling safe and welcome. He sleeps well when he doesn't feel like he has to protect himself. So just based on the way he sleeps, we can tell that Bud feels great with his new family in the jazz band. And remember how Bud's new band name is Sleepy LaBone? By giving Bud that name, it's as if Curtis is letting us know that from now on, Bud's got a real family, and he's going to be okay.