No, they cannot. If 7 years have passed since date of first delinquency on the original account (some say 7 years and 6 months), then you can have it deleted from your credit files due to obsolescence.
Do not acknowledge the debt is yours or agree to a payment plan or make a payment — any of these things could reset the statute of limitations allowing them to obtain a judgment in court.
Do not confuse your state’s SOL for legal action with the FCRA 7-year SOL for reporting adverse items on your credit files. These are separate deadlines.
You should consult an attorney, but it might not be worth it for a thousand dollar debt.