Around midnight six months ago, Time Warner Inc. CEO Jeff Bewkes received an expletive-laden email from a consumer. The man railed against his shoddy cable service and failed attempts by technicians to fix it, including missed service windows and excessive hole-drilling in his house. He called Mr. Bewkes a “f—ing piece of s—.”

Mr. Bewkes burst out laughing. He picked up the phone and called the man to compliment his writing style, but also to tell him he had directed his invective at the wrong guy, as many others have done....