At Glenmount, Dr. Mosley said, the school’s staff now collects phones at the beginning of the day and puts them in a phone locker. It has been “smooth sailing,” he said.

As an historic blizzard pummeled New York City Monday, about four dozen people dozed off in chairs in a lobby of Bellevue Hospital, slumped over suitcases or swaddled in blankets. Some were dressed in ...