No, he isn't going to sleep with all the fishies up in that big ocean in the sky. He's just resting ... I think. I'm not really sure what he's doing, but we often find him belly-up like this and think he's headed for the porcelain tomb. Then he just flips over and starts swimming again. It's weird and creepy.
Notice the low water level in the above photo. Harriette drinks poor Sharkbait's water. This is her right after we got him, caught with her paw in the cookie jar (literally!). Maybe that's why he plays dead ... "if I don't move, she'll never notice I'm here and she'll go away." Poor fish. He's just scared ... to death.