Ryan Collins of Aldie, Va., was only 10 weeks old when doctors made the diagnosis: Type 1 diabetes. That meant up to eight insulin shots per day, a big burden on him and his family. “He couldn’t be anywhere unless there was someone around to give a shot,” said his mother, Dana Collins. “Everything had to be planned. There was no impromptu anything.” Until last month, that is, when Ryan, now almost 7, stopped needing shots. Ryan, it turns out, does not have Type 1 diabetes after all. He has a rare form of diabetes, not yet discovered when he...