Since an "insurance eligibility point" is a New Jersey term, I will answer the question for a NJ resident. The answer to your question is yes, your insurance company can continue to assess you 4 insurance points even if they are no longer any motor vehicle violation points on your license. The information below explains why this is possible.

Whenever the NJ Division of Motor Vehicles assesses points on a driver's license for a moving violation, the same number of automobile insurance points are also assessed. These insurance points can allow an insurance surcharge to occur, thus a premium rate increase.

While traffic school and other means allow for motor vehicle violation points to be reduce or removed, insurance points cannot be take away in this same manner. The points that you had taken away helps keep the points total on your license from getting too high, but does not remove or reduce your insurance points. The violation itself also remains on your driving record.

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