I had a user send me this suggestion. I'm posting his solution in case someone else runs into this problem. Here is the text from his suggestion.
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Ok... seriously! You just gotta add this to the list of things to check. It's [stupid] and it has less than nothing to do with programming (and I would know, I'm a server engineer). Maybe programmers all just know this inherently, but while troubleshooting a client's server, we found that the SMTP service had been inadvertently assigned a specific IP address instead of using "(not assigned)". It was using 192.168.1.22 while the application was set to send mail to the service at 127.0.0.1. The application wouldn't send mail until we changed it back to "(not assigned)"

I know. Stupid, right?