Whether it's to deflect attention from their own situation, or an attempt to "get in" with a group to which they do not belong, or simply an attempt to shut up a message they don't like, or for some other reason, it's a common occurence.
This was recently brought home by the ironic situation of David Brooks of the New York Times claiming "Obama had a problem" because he wouldn't be comfortable with the people at an Applebee's. Except Brooks fucked up, showing he's completely ignorant of what can be found at an Applebee's himself.
This is not an occasional happenstance - the "charge" of elitism always seems to come from a member of some elite. It's hypocrisy writ large - writ large because the stakes are usually big.
They're making another person or group scapegoats for a "crime" that they themselves are guilty of, in order to incite support among a group of people to which they simply do not belong.
[And really, it's not such a surprise that this is almost always the case. An ordinary Joe doesn't use a word like elite. It's insufficiently exoteric (as are words like "insufficiently" and "exoteric" - the irony isn't lost on me, but at least I've linked to a definition - I may be an elitist bastard, but anyone's welcome to join me!)]
I'm almost tempted to call this a law:
(1) If a person calls another an elitist, they are almost guaranteed to be a member of a social, economic or political elite themselves.
If that law is true, it carries with it something like Godwin's law as a corollary -
(2) the first person to level a charge of "elitism" is a loser
The appropriate response is to simply indicate that we know the speaker is hypocrite - a loser of the worst kind. Let them know you're onto them. You know you want to.
I think this hypocrisy comes from a misplaced sense of self-importance - that the rules they play by don't apply to them. Helping deflate that sense of self-importance is a public service.
(Part II is here.)