ANSWER: The Bible, especially the Old Testament, does have quite a bit to say about debt (or going today what we would call bankrupt). 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 child of a widow could be sold as a slave to pay what their father owed (2Kings 4:1). The person who owed the debt, or his family, could be sold. It is also possible the person might have to become a slave for a period of time in order to pay back what was lent (see Leviticus 25:39, 47, Isaiah 50:1).
Regardless of how someone ended up in debt, those whom they owed were not allowed to oppression them but rather were commanded to treat them with mercy and leniency (Exodus 22:25 - 27, Deuteronomy 23:19 - 20, etc.). There are two principles regarding debt which, admittedly, are somewhat in tension. The first principle is that Christians should keep their promises and pay what they owe. This is part of the general rule of the ninth commandment which says we are not to bear false witness (Exodus 20:16). We could also use Numbers 30:2, which concerns vows, as a guide to lead us to keep our word and pay what we agreed to pay.
The second principle is that the lender should also be merciful and consider the plight of the person that has to go in 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 in the fiftieth year, which is stated in Deuteronomy 15, especially in verses 1 and 2.
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, God freed the slaves every seventh year who had entered slavery (at least often) because they could not pay what they owed.
Based on these principles, a person should strive to pay back any money they owe. If they cannot pay them back, such as meeting the minimum monthly payments on credit card bills even after cutting back at home (such as moving to a cheaper apartment or house, giving up a car, eating out much less, etc.), then it might be time to file for bankruptcy. Filing for this, however, should NOT be used as an instrument to escape your obligations through a scheme to defraud creditors.
In conclusion, please note that it is sometimes hard to determine from the Bible whether debt or bankruptcy is a sin. It depends, in part, on the circumstances. I would maintain that someone should still try to pay something on what they owe, especially when most credit card bills, etc., are NOT run up to put food on the table and to avoid the worst kind of deprivation (Deuteronomy 15). Instead, most debt is for luxuries and "wants," not "needs." Owning at least one or two pairs of shoes is a "need" but desiring ten or twenty pairs is obviously a "want."