Scott Kohan woke up in an Austin, Texas, emergency room after an attack that broke his jaw. The hospital was within his insurance network. But the oral surgeon who set his jaw wasn’t. Mr. Kohan’s insurer refused to pay the surgeon’s $8,000 bill. He’s not alone. An estimated 51% of ambulance rides, 22% of emergency-department trips, and 9% of elective cases, in which patients have time for due diligence, lead to surprise bills.