Adelphia.net account in Mail

Have you tried contact Adelphia for help.. You need an account to see the software settings necessary or access their support site, so....

I looked at a couple of sites on the Internet like www.adelphia.com/esafety/unsolicited_email.cfm and they indicate that the outgoing mail server should be 'mail.adelphia.net' not ' smtp.mail.adelphia.net'.

Are you certain that you are to use ' smtp.mail.adelphia.net'?

A Firewall being enabled shouldn't make any difference especially when accessing your Adelphia.net email account when at Home using Adelphia as your ISP for connecting to the internet and you should have the built-in Firewall included with OS X enabled for protection.

My outgoing server
is smtp.mail.adelphia.net

I use adelphia and that's not right. It should be

mail.adelphia.net

When you're at your office, you're not on your home network serviced by Adelphia–therefore using their outgoing server won't wok.
You need to see if you can use the ISP server for your office.
For example, when I go to school, I have to use my school's outgoing mail server. At home, I can use either my ISP (Road Runner) or my school account.