BERGAMO, Italy—Ambulances here have stopped using sirens. The frequent blaring only adds to local fears. Besides, there are few other vehicles on the road in Italy’s national lockdown.

Most are headed to the Papa Giovanni XXIII Hospital, a large, modern medical facility in a prosperous Italian city that has been overwhelmed by the coronavirus disease. There aren’t enough ventilators to intubate all patients with Covid-19 who have severe breathing trouble. The intensive-care unit is taking almost no patients older than 70, doctors said.

A normally disused section of the hospital is filled with the critically ill and the hissing sound of oxygen. Patients lie quietly, with worried or exhausted faces, visible to others in the series of half-open rooms. Each focuses on the struggle to breathe. There are patients with airtight oxygen helmets over their heads, like transparent buckets taped at the neck.

“Some of them would have needed intubation in intensive care,” anesthesiologist Pietro Brambillasca said. The rest ought to be better isolated, he said, where they can’t contaminate anyone.

That is no longer possible. The number of ill has outstripped the hospital’s capacity to provide the best care for all.