"Sorry for being three minutes late."
As an example of the apology-that-isn't-quite, it was really rather elegant. First there was the omission of that inconvenient little word "I." Nothing so direct and personal as "I'm sorry," or even "I was late." No subject in the sentence at all. Just the breezy, impersonal "sorry" that implies a certain level of mild regret on a vaguely global scale without necessarily acknowledging that the speaker personally had anything to do with whatever may or may not have happened.
The master stroke, however, was the speaker's subtle but unmistakable emphasis on "three minutes" rather than "sorry." This was a delicate but oh-so-clear statement that any transgression that may have inadvertently taken place was so minor and insignificant that no one could possibly be upset by it unless that person were an unreasonable, obnoxious jerk.
This neatly preempted any possible complaints from the five of us who had been standing outside the locked door of the fitness center wondering why the door was still locked when it was after opening time. If we expressed any annoyance, we would be obnoxious jerks. Especially if we were so unreasonable as to point out that we had, in fact, been waiting for longer than three minutes.
This placing the responsibility on the apologees rather than the apologizer was skillfully done. It was almost up there with the classic phrasing from the public figure who really isn't sorry at all: "I regret it if anyone was offended."
I doubt whether those of us who had been waiting were particularly annoyed at the employee's minor tardiness. Certainly no one said so. We knew perfectly well that any of us can be rushed, forget to watch the time, mislay our keys, or for all sorts of other reasons end up keeping people waiting. It's an ordinary, understandable, and forgivable thing to do.
But any of us can also say, simply and sincerely, "I'm sorry." A genuine apology instead of a pseudo-apology builds connections with people instead of brushing them aside. When we take responsibility for our own little human errors, we make it easy for other people to forgive us, because we are treating them with respect. Most of the time, we'll get their respect and quick forgiveness in return.
But even if it doesn't persuade others to forgive us, there's another reason why a genuine apology is a good idea. It just might keep people from posting snarky little rants about us on the Internet.