IANAL, nor an insurance expert. I’ve just had people drive into my car for no apparent reason several times, and also had a person pull out in front of me when she had a stop sign. The following is based on those experiences.
NJ is a “no fault” state. Your own auto insurance pays all bills as submitted minus whatever deductible you have chosen. Period. If the insurance co believes the injuries were someone else’s fault they may go after that person and his/her insurance company for the total amount paid. If they are successful, you may get some or all of your deductible returned to you. This is called “abrogation.”
If the injury is sufficiently severe – for example, if you have broken bones – you can also make a claim against the responsible party for pain and suffering. Note that bruises and whiplash, although they can be extremely painful and cause suffering, are usually insufficient for such a claim. At least that’s what the lawyers told me.