This is a question better suited for a bankruptcy attorney.
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If you were employed directly by the military and not through a contract agency you would not be able to discharge the debt as it is owed to the government. However, if you get a job you could repay it through chapter 13 over a period of time.
I have to admit, I have not dealt with or seen this situation. For a debt to be nondischargeable, it has to fit within one of the categories set forth in Sect. 523 of the Bankruptcy Code. The list includes taxes, student loans, child support, debt caused by fraud, etc. You did not defraud anyone and bankruptcy is supposed to help those who are thrown into unfortunate circumstances. I think the government would have a hard time winning on this one, but there will be procedures to find out for sure.
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In my experience the answer is no. If you call DFAS, they will work out a payment plan for you or they will start attaching your IRS tax refunds. Further, they will turn it over to a collection agency. A competent bankruptcy, particularly one near a base, almost certainly can answer this off the top of his or her head. If there is an amount in controversy - you may want to ask for an audit of your military pay, e.g. you should only be paying back a pro rata amount - i.e. if you signed for 4 years, got a bonus, you may only owe 75%.
I did have a case similar to this. A few years ago. My client had received a bonus along with a relocation payment and when he got to the new base he didn't pass the physical. He was honorably discharged but eventually filed bankruptcy in part due to the collection actions regarding these payments. We were "kind of" able to discharge the debt. It was the first time I had encountered this but if I recall, it's treated similarly to an overpayment of V.A. benefits in that we were able to stop collection of the debt but if the debtor ever became eligible for benefits, the debt would be recouped from future benefits. I hope this helps, good luck.