The answer depends on the nature of the actual order. A judge does not order damages as part of a contempt action. A contempt may be either criminal or civil in nature. A fine is also criminal in nature and thus is not stayed by the bankruptcy court
Whereas a civil contempt order requiring you to do something that you were already ordered to do, for example, to pay the other party's attorney's fees at a prior hearing, could be stayed by the bankruptcy court.
It's a very complicated issue. In general, however, filing bankruptcy so as to avoid a contempt order is probably not very cost effective. Especially with a Chapter 13, you would be effectively putting all of your disposable income at risk for the next five years. In the same amount of time, you could probably satisfy the contempt order without filing bankruptcy.
If you want to explore the possible outcomes, you may want to have your contempt order reviewed by a local bankruptcy lawyer. Most lawyers give free first consultations in order to determine whether or not a bankruptcy is the right thing in your circumstances.
For a referral, see this link.
Hope this helps.
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