i run into this a lot at work, where we'll be collaborating on something, but i need to put aside messages from my a friend or family member until i have a moment alone. While another developer is looking over my shoulder (pair-programming-ish), I need to know that there's a message pending for me, but I have no way of knowing if it's sensitive content. I may want to put off reading the message until I have a moment alone, or I may want to dismiss the person that can see my monitor for a few moments.

In this case, we want to know that a message came in (possibly knowing who it came from), and that it should be checked when there's nobody looking over your shoulder. Under gnome-shell, the message pops into the messaging bar if your status is non-busy, or "into the abyss" if you're busy. (an "urgent" hint may be displayed somewhere in the overlay if you happen to open it eventually)

To be clear: it's not really a matter of being too busy to deal with it, it's a matter of privacy. "busy" is really the wrong state here. you're not busy, you're just not in a safe state to recieve potentially sensitive messages.