THE show business adage that “there’s no such thing as bad publicity” is probably true about 99% of the time. But as the media — and its consumers — grow obsessively focused on the personal struggles of Hollywood stars, there are rare instances when even a little free exposure can be problematic.

On Friday, Paramount Pictures will release “Drillbit Taylor,” a new comedy starring Owen Wilson as a bodyguard hired by several high school students looking for bully protection. The film ...