They can do this if you do not assert the affirmative defense of the debt being older than the statute of limitations. I can't opine on AZ law, but it appears from online sources that the statute of limitations for credit card debts there is 6 years, so if nothing tolled or re-started the running of the SOL clock, you should not have a problem. In some states, making even a small payment on a debt can re-start the clock. The problem in those states can be the "phantom payment" ... they say you sent 'em $50 four years ago and the judge may decide that you need to prove otherwise or the SOL defense won't be accepted. There have even been cases where phantom payments were actual payments, but not made by the debtor: the collector was walking next door to the 7-11 and buying small money orders.
I am not admitted to practice in AZ but the account appears to be time barred. You need to confirm this with an experienced AZ consumer lawyer. If the account is time barred, you may want to sue the debt collector for threatening to sue you on a time barred account.
If you email a message containing my response along with your name, I believe I know at least one AZ attorney who could assist provide you with legal advice whether you have an FDCPA claim worth pursuing.