You requested that the other account holder close the account, and never followed up? This is not a unique story.
The statute of limitations is a bar to collection through the courts. This sounds like a collection as a matter of contract, probably both through the original account and through your new account, with a clause permitting the back to deduct any overdrafts or fees owed from current balances. (Check the contract.)
Also, even if the statute of limitations were relevant, it would likely be the ten year Illinois statute governing written contracts.

