How long does a credit card company have to sue?

It varies from state-to-state and it also varies as to the type of debt in question (e.g. credit card debt, oral contract, etc.). You can google your state's "statute of limitations" as to the appropriate timeframe in your jurisdiction. Once the stautue has run, a creditor can still try to collect on the debt (technically you still owe it), however it can no longer take you to court over it.