About 28 million Americans have sleep apnea, which causes repeated awakenings and pauses in breathing during the night, sometimes resulting in loud snoring and gasps for air. For decades, the standard treatment has been “continuous positive airway pressure.” A mask worn at night pushes air into the nasal passages, enabling easier breathing. C.P.A.P. reduces and in some cases completely prevents episodes of apnea. But the mask is like something from a bad science fiction movie: big, bulky and obtrusive. Many patients simply refuse to wear it or rip it off while asleep. Studies show that about half of all people...