Jim,
Private messages are very quick. The messages stay in your outbox until the user they are sent to checks his/her messages and then are moved to their inbox. If the other user has a popup blocker running, there is a very good chance they will not see that they have a new private message since no popup will appear on their machine informing them of a new message. The only way they will get the message is if they notice the new message link at the top of the page next to the logout link.