Stacy Jorgensen fought her way through pancreatic cancer. But her struggle was just beginning.

Before she became ill, Ms. Jorgensen took out $43,000 in student loans. As her payments piled up along with medical bills, she took the unusual step of filing for bankruptcy, requiring legal proof of “undue hardship.”

The agency charged with monitoring such bankruptcy declarations, a nonprofit with an exclusive government agreement, argued that Ms. Jorgensen did not qualify and should pay in full, dismissing her concerns about the cancer’s return.

“The mere possibility of recurrence is not enough,” a lawyer representing the agency said. “Survival rates for younger patients tend to be higher,” another wrote, citing a study presented in court.

There is $1 trillion in federal student debt today, and the possibility of default on those taxpayer-backed loans poses an acute risk to the economy’s recovery. Congress, faced with troubling default rates in the past, has made it especially hard for borrowers to get bankruptcy relief for student loans, and so only some hundreds try every year. And while there has been attention to aggressive student debt collectors hired by the federal government, the organization pursuing Ms. Jorgensen does something else: it brings legal challenges to those few who are desperate enough to seek bankruptcy relief.