Considering the plight of most people in her situation, Becky Hemingway was faring well. Her right side had gone numb, and she couldn’t speak. But her family had immediately suspected a stroke, and emergency room physicians at a hospital in South-bridge, Mass., not only made the diagnosis but also rushed the 31-year-old new mother to a regional stroke center in Boston, two hours to the east, where she could receive specialized care.

In contrast, the average stroke victim doesn’t get treated for almost a day…