Since the last payment was in 2013, the statute of limitations wouldn't apply (i.e., they have 6 years to file a lawsuit).
You can certainly argue that you never took out the credit card. But my suggestion would be to immediately file a report of identity theft with the police. If you do not, then your claim that you didn't open the account would probably not be very believable. If the police investigate and there seems to be evidence that it is not your account, then that is certainly a viable defense. Of course, all of that would be pointless if they have a signed agreement from you, or if they can show that you made payments (i.e., canceled checks, or money transfers from your bank account). Also, bear in mind that it is a crime to falsely report to the police that your identity was stolen. Accordingly, you will need to be sure of what you're doing.
If you know that you cannot win with an identity theft argument, then my suggestion would be to attempt a settlement. This could involve a lump sum payment for less than what is owed, or a payment plan, or both. Ultimately, if the amount owed is just too large to deal with, bankruptcy is an option.
I know this isn't the type of answer that you were hoping for, but under the circumstances I hope you understand why this is the answer, and how best to proceed.
Please let me know if you need clarification, as I am happy to continue helping you until you are satisfied.