If you have a broken ankle, do you sleep in the boot?

Cast or "the boot" "boot" meaning the fracture walking boot - usually yes, as the boot is a hi-tech version of the plain old cast used to prevent from the fracture to displace and loose its proper alignment.
Maybe. Some physicians say yes, keep immobilized at all times and others say its okay to remove boot at night. It usually depends on the severity of the injury. Checking with your doctor is the best thing to do.