In New Jersey if you fail your inspection, you will be allowed a specified amount of time to have the necessary repairs completed and your vehicle re-inspected. You can either have the repairs completed by a registered Emissions Repair Facility (ERF) or you can make them yourself. When you fail the inspection, you will get an Emissions Repair Form and a Vehicle Inspection Report that will help determine what caused the emissions problem.
New Jersey Administrative Code 13:20-43.12 speaks about inspection extensions. Here it states that the owner of a motor vehicle registered in NJ which has failed inspection and requires repairs, other than repairs required to be made pursuant to N.J.A.C. 13:20-7.6 (immediate repairs needed due to hazard), which cannot be completed prior to the date by which the motor vehicle must be presented for re-inspection due to the nature of the repairs which are required, shall notify the Division of the date upon which the repairs to the motor vehicle shall be completed.
If a current registration has been issued for the motor vehicle, the inspection Director may issue a letter extending the time period in which to have the motor vehicle re-inspected. The inspection extension shall be valid from its effective date to 14 days from the date upon which the repairs to the motor vehicle have been completed, but in no event shall the extension be valid beyond the expiration of the two year inspection cycle established for the motor vehicle.
From the NJAC information above it could be possible that the expiration of your inspection cycle has been passed and that is what the officer was referring to and the reason he was giving that the car could not be driven without receiving a ticket in the future.
If you know this is not the case then I would suggest calling the 1-888-656-6867, the number that the New Jersey Motor Vehicle Commission (MVC) gives for questions regarding inspections. Here they should be able to tell you for certain if you can drive your car with your extension or if you could get a ticket like the police officer indicated to your son.