No, refusing a counter offer would not be "bad faith."  Bad faith is when an insurance company refuses to pay a legitimate claim of its own insured.  You can not bring a bad faith claim against someone else's insurance company.  There is no responsibility for an insurance company to respond to every counter offer with another offer of its own.  Say, for example, an insurance company offered $1,000.00 to settle a claim, and the accident victim countered with a demand for $1 million.  The insurance company is going to stop negotiating because the demand is way too high.  That is not bad faith.