Dr. Christopher Lucas shuttled from room to room, checking on the children with mental-health troubles who had streamed into his emergency department over the past 12 hours because they had nowhere else to go.

There were eight of them that September day at Upstate University Hospital in Syracuse, N.Y. In one room, staff tended to a 17-year-old girl with chronic depression who had attempted suicide by overdosing on ibuprofen—her fourth trip to the E.R. for mental-health reasons in two years. Nearby was a 14-year-old girl who had started cutting herself after being bullied over social media.