Non-friends on Facebook aren't getting my messages?

[JZStudent]

I've been sending a couple messages to people I know, but aren't added as friends on Facebook. However, I don't know if they even received them because I tested sending messages to other accounts I own to test out features on Facebook, only one of the accounts got the message, and the other didn't. I was really confused why. The account sending the messages is brand new, and has no friends. So I sent a message to two other accounts but only one got a message. Is this an error or will the message ever show up? The messages I sent out to the non-friends are very important so I need to know ASAP.