When Claire, a pixie-faced 6-year-old in a school uniform, heard her older brother, James, enter the family’s Manhattan apartment, she shut her bedroom door and began barricading it so swiftly and methodically that at first I didn’t understand what she was doing. She slid a basket of toys in front of the closed door, then added a wagon and a stroller laden with dolls. She hugged a small stuffed Pegasus to her chest. “Pega always protects me,” she said softly. “Pega, guard the door.” James, then 10, had been given a diagnosis of bipolar disorder two years earlier. He was...