After moving our phpbb3 installation from a shared hosting account to a VPS account with 1&1 all of a sudden our mail notifications were not working. We do not need a full blown mail server on our VPS so we still wanted to use the regular 1&1 SMTP servers to send out the mail for the board. After trading a few emails to 1&1 support we got it figured out. When we moved our domains over to the new VPS package we took one of our original accounts that was the account we were using for SMTP and made it a mail forward only. Once we gave that account an actually mailbox and password it worked.
Two things that are odd, one that it seems the mail does not send as quickly from the VPS package instead of the shared hosting, and the second one is that I tried to change the SMTP information in phpbb3 board to an account that had a real mailbox but it never worked. Also for some reason sending yourself test PM messages didn’t work, does the phpbb3 board know that you are sending them to yourself and then not allow you to get a notification?
Also I could never get the gmail SMTP stuff to work correctly even though I tried following what other people said.