Yes, it may be possible, but it is not easy. In a recent case in Houston, a bankruptcy judge ruled that a debtor is not required to prove the existence of a completely debilitating medical condition in order to establish "undue hardship" under the Bankruptcy Code, so that a student loan borrower could discharge his loan.
If a debtor can demonstrate that some condition will, in all likelihood, inhibit the longterm ability to repay the student loan debt, at least one element necessary to discharge the debt has been satisified. Moreover, a debtor is not required to present expert testimony to corroborate his own testimony about his health. But it is not an easy burden; you have to be so ill that you are basically unemployable.
Remember, if you are totally disabled, you can seek to have your student loans forgiven by the Department of Education under a "disability discharge," but they must be federal student loans. Private student loans usually do not provide for disability discharge.