You don't realize you're playing this game until the other person takes their turn - one-upping whatever self-deprecating observation or grief-airing you've just expressed. No matter how much more you stretch your own terrible thing, the thing they're talking about is much worse, and they'll interrupt or talk over you to emphasize how much worse the thing is - even if it's their own flaw they're talking about.
Fortunately, being aware of the game makes it a little more easy to avoid forcing others to play it. I try to not one-up people. There's a thin line between sharing related anecdotes and one-upping, though, and I'm a little paranoid about falling on the wrong side.