Assuming that you are referring to liability insurance, the general answer, subject to the law of the jurisdiction, is "Yes". This presumes the legal, provable damages exceed the policy limits, rather than a claim made at the "I want" stage of things.
A factor that is important to remember is that a liability insurer has an obligation to resolve a claim on behalf of an insured within the policy limits if it is all possible to do so. This generally translates into a scenario where, if the claimed damages may reasonably be expected to exceed policy limits (for example, if the case goes to a jury), but the claimant is willing to accept policy limits in full settlement and release the insured from further liability, the insurer has an obligation to settle within limits. By so doing, it is protecting the insured from individual liability for an "excess" verdict, and is fulfilling its duty of "good faith".
Many States have statutes (written laws) that provide a process by which an insurer may be held liable for an excess verdict. If it is shown that the insurer did not act in good faith by settling within policy limits when it had the opportunity to do so, the insurer may be held responsible for amounts in excess of its policy limits. Those amounts may include the full amount of the verdict, damages to the insured that proximately flowed from having the excess verdict/judgment entered (such as damage to credit), attorney's fees, and depending upon the jurisdiction, other categories of damages.
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