Virginia Murphy borrowed a small fortune to attend law school and pursue her dream of becoming a public defender. Now the Florida resident is among an expanding breed of American borrower: those who owe at least $100,000 in student debt but have no expectation of paying it back.

Ms. Murphy pays just $330 a month—less than the interest on her $256,000 balance—under a federal income-based repayment program that has become one of the nation’s fastest-growing entitlements. She plans to use another federal program to have her...