Maybe a little . . . sometimes.
Not satisfied with that answer? You shouldn't be! General liability insurance has sometimes responded to privacy incidents under its Personal & Advertising Liability coverage, but this response is predicated on the release of information being considered publishing of that information. That interpretation has been unpredictable at best.
Some other policies (e.g. professional liability) may respond, or may even have been enhanced to respond, but, again, that's not reliable. Much of the value in cyber insurance comes from response services. You want an insurance company to respond purposefully and expertly; and that's only going to happen with a true cyber insurance policy.