Scientists believe that if whales and dolphins do dream, it is for very short spurts at a time. When they sleep, only half of their brain shuts down, while the other half remains alert enough to be aware of predators and other threats. This state is likley similar to that twilight period we experience between waking and sleeping. During sleep they will also keep one eye open.  After about 2 hours they will reverse the process, shutting down the other half of the brain and opening the other eye.  Dolphins often sleep in pairs, and the one eye that each keeps open will be facing the other. 
While our breathing is regulated by the subconscious mind – we don’t have to think about it – their breathing is actually a conscious process, and they must stay at least partially awake to ensure they will breathe often enough. They also are usually swimming at least to some degree while they are sleeping, and during R.E.M. (dreaming) sleep the body becomes paralyzed. So dreaming for any extended period of time would not be conducive to the survival of these animals.
What about sleep talking, or learning during sleep? Researchers in France recently recorded some dolphins talking in their sleep! 5 dolphins were born in captivity and live in an aquarium where they put on shows every day. One day the show’s theme music included songs of the humpback whale. About a month later the researchers noticed that while the dolphins were sleeping, they started making whale sounds. When the recordings were played back for biologists, many of them thought they were listening to whales, not dolphins. I guess these intelligent creatures are bi-lingual!