Depends on the amount. It is not free to post things to credit bureaus. If it is a small amount or not a strong claim, they can and do give up. Credit cards are considered an extremely strong claim, cancellation fees are not for example.
Credit cards are charged off by the lender for most people around D90 and then sold. The debts that are purchased are given to secondary companies that try to collect. These debts can and are sold several more times for fractions of a fraction of a penny to increasingly aggressive collectors, but even they run out of steam.
I won a fight with Direc TV who claimed I owed them 400 dollars simply by doing nothing- their threats were empty but did accelerate until they stopped completely at around the 5 month mark. I answered the harassing calls and explained my situation. Once the debt was sold, I just wasted their time since they no longer have any power to eliminate the debt outright but they will wheel and deal since they bought the debt at a % of what is outstanding. However, make sure they send you WRITTEN TERMS that say you'll settle for X amount. I almost did this with Direc TV to get it dealt with but was told that they'll just lie to you and continue following up on the debt.
If you ever get a very old debt notice in the mail, never pay anything towards it. It will run off on its own at around 7 years on your report, but acknowledging the debt can reset the clock.