In more than 40 years as an animal rescuer, Bill Caprio helped countless dogs, a duck, a pig, even a rooster. Oh, and also a baboon, some foxes, the occasional hedgehog and a boa constrictor. But one of his most memorable cases involved Sinbad the cat. A meter-reader had called in the report. “The cat looked like it was dragging a carpet,” said Elliott Serrano, an education manager with the Anti-Cruelty Society.

It was December 2016, and Mr. Caprio was dispatched to Sinbad’s home. He discovered the Persian mix dragging five pounds of matted fur, along with dirt and maggots. Colleagues said Mr. Caprio used diplomacy to convince the elderly, financially pressed owner that Sinbad would have a better life if he gave up his pet.