(idiomatic) A situation in which a person unsuccessfully attempts to contact another person by telephone and leaves a message instead, and in which the second person then unsuccessfully attempts to return the initial call and leaves a message for the first person, and so on as if the two are playing a game of tag in which the most recent person to have been left with a message is now designated as "it" (i.e. as the player now obliged to chase the other and to attempt anew to make contact).
Playing telephone tag with your doctor is one of the most common—and frustrating—rituals of modern medicine. You phone...can't get past the receptionist...leave a message. Your doctor calls back while you're in a meeting...leaves a message. The amount of time wasted on both sides turns everybody sour.