The Bible, especially the Old Testament, does have quite a bit to say about debt and bankruptcy. It was generally looked upon as being a misfortune, brought on by things such as a bad crop or the death of the family's breadwinner.
A person who was in debt and did not repay it could find himself or herself as someone else's slave for a period of time in order to pay back what was borrowed (Leviticus 25:39, 47, Isaiah 50:1). A child of a widow could be sold as a slave to pay what their father owed (2Kings 4:1).
Regardless of how someone ended up being a debtor, those whom were owed were not allowed to oppression them. They were to be merciful and lenient to those they lent money or goods (Exodus 22:25 - 27, Deuteronomy 23:19 - 20).
There are two principles regarding owing others that, admittedly, are somewhat in tension. The first principle is that Christians should keep their promises and pay what they owe. The basis for this tenet is the ninth commandment which states we are not to bear false witness (Exodus 20:16). We could also use Numbers 30:2, which concerns vows, as a guide in keeping our word. The second principle is that the lender should also be merciful and consider the plight of the person that has to go into debt.
The Lord is well aware that some poor people will not be able to pay back money loaned to them. The loans they get are really gifts by the creditor to aid them in their poverty. Notice the law concerning what you owed was completely cancelled every seventh year in a seven-year cycle that lead up to the Jubilee year, which is stated in Deuteronomy 15.
Interestingly, not only did God say that interest should NOT be charged on a debt, he forbad the lender from keeping things necessary for living even when the borrower offered such as security for a loan. Exodus 22 states that a lender was not even to take a person's coat as a pledge of repayment but rather give it back to him (Exodus 22:25 - 27).
God recognizes how people in poverty can be so buried in debt that they cannot ever be freed by their own efforts. Similarly, he freed the slaves every seventh year who had entered slavery because they could not pay what they owed.
Based on these principles, a person in debt should strive to pay back any money they owe. If they cannot pay them back, and are even unable to make regular payments to the lender (even after cutting back their expenses), then it might be time to file for bankruptcy. Filing for this, however, should not be used as an instrument to escape obligations through a scheme to defraud creditors.
In conclusion, it should be noted that it is sometimes hard to determine from the Bible whether a particular debt or bankruptcy is a sin. It depends, in part, on the circumstances. A person should always try to pay something on what they owe, especially when most credit card bills are not high due to trying to put food on the table and avoid the worst kind of deprivation (Deuteronomy 15). Instead, most of what many of us owe is for luxuries and "wants," not "needs."