It sure seems like it. There are many examples of hard-wired human instincts that help keep us alive. Perhaps the most obvious case is the fight-or-flight response, coined by Harvard University physiologist Walter Cannon in 1915.
There is no time limit on that debt. A collection agency can legally pursue you until the debt is paid in full.   However, there is a time limit on the negative consequences of not paying the debt. In most circumstances that limit is seven years. Seven years from the time you became delinquent with your original creditor.
If so, the credit card (CC) holder may have to pay a credit limit fee for overreaching their limit. At the end of the month, the CC holder can choose to pay the entire amount back or leave an unpaid balance that will continue to accrue high interest until paid off unless it has a zero introductory APR.