Showing that your student loans present an undue hardship is nearly impossible in practice.
This Brunner test adopted a standard for "undue hardship" requiring a three-part showing: (1) that the debtor cannot maintain, based on current income and expenses, a "minimal" standard of living for herself and her dependents if forced to repay the loans; (2) that additional circumstances exist indicating that this state of affairs is likely to persist for a significant portion of the repayment period of the student loans; and (3) that the debtor has made good faith efforts to repay the loans. In re Brunner, 46 B.R.752, 756 (S.D.N.Y. 1985).
Mark Tetzlaff, who filed a Chapter 7 Bankruptcy in 2012, was asking the court to use a less strict standard to determine the dischargeability of his student loans. This test, known as "The Totality of Circumstances Test" and is arguably an easier test than the Brunner test.
Either way, the Supreme Court refused to hear his appeal, leaving the issue partially unresolved. For Mr. Tetzlaff, many will argue that he can still enter into a low Income Based Repayment (your IBR payment could be zero in some cases), leaving him with the ability to discharge his student loan debt over time, anyway.
Read more about it here.