MILAN—In the town of Coccaglio, an hour’s drive east of here, the local nursing home lost over a third of its residents in March. None of the 24 people who died there were tested for the new coronavirus. Nor were the 38 people who died in another nursing home in the nearby town of Lodi.

These aren’t isolated incidents. Italy’s official death toll from the virus stands at 13,155, the most of any country in the world. But that number tells only part of the story because many people who die from the virus don’t make it to the...