Cyndee Marcoux already was stretched thin, thanks to the $80,000 in student loans she racked up after getting divorced and going back to school a decade ago. Her breaking point came in 2010, when her daughter defaulted on student-loan payments of her own.

That's because Ms. Marcoux, a 53-year-old library administrator in Seekonk, Mass., co-signed for about $55,000 of her daughter's loans. When the daughter was unable to keep making payments, Ms. Marcoux was on the hook—a burden that forced her to reshuffle...