I had some PM's that I had received recently sitting in my inbox that I no longer needed, so I decided to go ahead and delete them. After I deleted them, I got emails stating that I had received private messages (the same ones that I deleted).
I hadn't received the notice in the past, only got it today when I deleted them.
As a side note, there's no indication on the site itself when you receive a private message either. Usually there's a pop-up or number indicator next to "Private Messages" or "My Account", but I haven't seen anything.