Hillary Clinton recently drew scrutiny for a claim that resurfaced in a New Hampshire breakfast on Tuesday that she once tried to join the Marines. The dubious claim, which even implicitly accused the Marines of sexism, brings to mind another case of military optics that might not have worked out the way Mrs. Clinton wanted it: A photo of her taken while shaking the hand of a soldier. The photo inevitably made the rounds in chain emails and on websites for years. But unlike some stories that seem too good to be true, and frequently turn out to be false,...