It takes me an unusually long time for me to send PM’s, once even to the point that I thought I couldn’t send any, period.
For instance, I’ve submitted a PM for one of the administrators (the topic is not important) to my outbox, and it’s just stayed there for over 3 hours.
The only cause I can possibly think of is the fact that I don’t like removing old PM’s. Any suggestions? 
If the PM is in your ‘Outbox’ this means that the PM has been sent, but the other person hasn’t read it yet. Once you see the message has been moved to your ‘Sentbox’ this means that the other person has read the message. Hope this clears things up for you. 
Maybe that administor hasn’t read it yet or doesn’t know that he/she has a new PM. I’ll suggest to be patient. After a while, try PMing him again.
Thank you for the clarification about the Outbox.
You’re welcome!