Pay particular attention to the settings for IP Address Restriction and Configure Authentication. If you have IP address restrictions in place, per the article, you'll need to make sure they match up with the machines that are having trouble sending. If you have authentication configured, you'll need to make sure the sending clients are actually authenticating.
Barring these two suggestions, I recommend using the telnet command to test sending a message via POP and see what error message the server is sending back to you. While this article isn't specifically about Exchange 2003, the steps for telnet described in it can be used to test an Exchange 2003 server: Knowledge Base article 885685: How to troubleshoot the POP3 Connector in Windows Small Business Server 2003.
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