The neighbor's insurance company does not owe you a duty of good faith. It is your insurance company that has a duty of good faith toward you. You should make the claim to your own insurance company and let it worry about getting its money back through subrogation. If your neighbor was not negligent or no negligence can be proven, they are denying liability, not denying coverage. If you sued the neighbor, their insurance would provide a defense and cover them. You should have homeowner's insurance. Make the claim to your insurer.
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You cannot sue your neighbor's insurance company for bad faith. A bad faith cause of action lies only against your own insurance company. Have you submitted this to your insurance carrier? Your insurance carrier should've picked up the claim and then pursued your neighbor's insurance carrier, in subrogation, to Recover what they pay you. I specialize in insurance coverage. If you would like, I would be happy to provide you with some free guidance. Give me a call after the holiday.