You can certainly sue your neighbor. Insurance is simply a "source of payment" if you prove damages. You need to balance a number of factors, including cost, your time, and your relations with your neighbor. Best of luck.
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You can present a claim through your own carrier for the damage and rental (if you have that coverage) to avoid the delay and hassle. The other option is to sue the driver that hit your car. As indicated by others, you will need to consider the costs and time involved with your options.
other than going under your own insurance policy (collision coverage - if you have it), your only alternative is to file suit. "Assigned Risk" carriers are an abomination and will generally do what ever, and for however long they can, to avoid paying out. If your damage is $5000 or less, file a small claims action. If its more, consult with an attorney in your area.