Unfamiliar Acquaintances

Every so often, we run into people we recognize but can't quite place. Perhaps you catch sight of a strangely familiar face at your favorite coffee shop, and then later at a diner while visiting family out of town, and are puzzled by the coincidence. Write a story in which two of your characters keep crossing paths, either accidentally or because of particular circumstances. What keeps them from properly introducing themselves? Could they become good friends, or will they become adversaries?