My first thought would be that your mail is being intercepted by a spam filter but you have not provided much information about your ISP or how your email account(s) are set up. I have assumed we are talking POP.
There is nothing wrong with Mac OS X 10.3.9 or Mail.app that would cause such a problem.
Do you have more than one email address (not email alias)? If so, send your email to another email address and see if that turns up. If you don't have another email address then create one (Yahoo or Gmail) and check webmail to see if it is reaching there. Then set up to have it received in Mail.app and see if it turns up.
I'll bet there are at least 50 guys out there that can straighten out my Apple iBook g3 laptop OSX 10.39, Mail 1.3.9 in a minute. It might be the machine they learned computers on 10 years ago. I can sit here and send out 100 emails that'll swoosh out like rockets. When I address one to myself, it disappears into the ether never to be seen again. There you you have it: great sender, rotten receiver. No idea is too cockamamie, tell me the first thing that comes into your mind as to what I should check. I've made no additions, subtractions, or adjustment that I can remember. C'mon, get me straight.