Any creditor can legally sue you for failure to pay the bills on time. This is because you have a legal contract and obligation to repay the debt as agreed, and failure to do so is a violation of civil law. Honestly incurring and not repaying a debt cannot lead to criminal consequences, however.
You would be served with a summons if your credit card company or bank wishes to sue you in court. If you do not appear as requested, the creditor will automatically win the lawsuit against you. If you do appear, you may be able to work the situation out with the creditor and the judge.
Once a judgment is entered against you, it lands on your credit record as a serious public record entry. The judge can order your assets or salary garnished to repay the debt claim and lawsuit. Judgments usually include legal fees, interest, court costs, and other penalties.
If you are not working, the creditor will not be able to pursue garnishment of your wages. However, an unpaid judgment can be indefinitely renewed in most states and thus will remain on your credit report. Once a judgment is paid, it should fall off your credit files after 7 years.