Mail Test
Having had to do this recently, here is a good article on how to test for SMTP connectivity (which also allows you to test for open relays and send email from telnet)
Tech-Recipes - Send mail via telnet / test for open relay

But this is only covers SMTP connectivity.

To test IMAP, follow these instructions:
- telnet 143, the server should answer: * OK Server Ready
- 1 login , the server should answer: 1 OK LOGIN completed
- 1 list "*" "*", the server should answer with the list of directories for the account and terminate with: 1 OK LIST completed
- 1 logout, the server should answer with : * BYE Server Disconnect and 1 OK LOGOUT completed