The credit card company can continue to try to collect what you owe to it indefinitely. However, the statute of limitations on credit card debt, which is what I think you are referring to in your question is 3 years in Oklahoma. The statute of limitations limits how long you can be sued for the debt. In other words, once the statute of limitations on your debt has passed, you cannot be sued for the money you owe. However the credit card company is still entitled to its money and you are still legally obligated to pay it. Therefore, the credit card company may continue to try to collect from you itself, may turn your debt over to a debt collector, or may sell your debt to a debt collector. Meanwhile, the fact that you owe the debt will continue to harm your credit histories and bring down your FICO scores.
If you cannot afford to pay the full amount of the debt, I suggest you talk with a reputable debt negotiation firm or with a bankruptcy attorney and try to resolve your financial situation using one of those options. Click here to a free consultation with a bankruptcy attorney. I also encourage you to learn more about debt negotiation by listening to this debt settlement podcast.