An insurance policy is a contract. When a covered claim is made, the insurer must handle it according to the policy and the law that applies.
Fair claim handling
Depending on the jurisdiction and policy, an insurer may be required to investigate reasonably, communicate its position, evaluate relevant evidence, and avoid unreasonable delay, denial, or underpayment.
Your insurer and the other insurer are different
Your own insurance company owes contractual obligations to you. The at-fault party's insurer generally protects its insured and does not owe you the same contractual duties. That distinction matters when evaluating a possible bad-faith claim.
This article provides general information, not legal advice. Laws and deadlines vary by jurisdiction and facts.