A public cafe that combines the cat-loving cappuccino set with slumbering felines. What could go wrong?

A lot, it turns out: ill-timed belly rubs, great escapes, protests from animal rights activists and reluctant landlords claiming to be “dog people.”

These are the travails faced by entrepreneurs around the country who have opened more than 40 “cat cafes.” It can be a frisky business.

“I wasn’t ready for all the things that can go wrong,” says Sana Hamelin, owner of the Denver Cat Co.

Ms. Hamelin once lost a cat inside the 1,500-square-foot cafe for two weeks, although “Gus” crept out at night to nibble food. A cat named Morpheus bit a hands-on patron. “You shouldn’t go for the belly rub,” Ms. Hamelin says. “The belly rub is dangerous.”