Private Student Loan Bankruptcy Fairness Act of 2013 - Amends the federal bankruptcy code to limit the non-dischargeability, except in cases of undue hardship, of educational loans to those made, insured, or guaranteed by a governmental unit, or made under any program funded by a governmental unit or any program for which substantially all of the funds are provided by a nonprofit institution (thus allowing the discharge of private educational loan indebtedness without the need to show an undue hardship).