Administering two imap accounts with gnus?

[Hermann]

Hi,
recently I've set up a mail account at the German provider web.de. This
is not my main account, it is at my ISP 1und1.
I want to administer both accounts via nnimap in Gnus, and with 1und1 it
is no problem.
However I can't manage to get web.de working:
In ~/.authinfo there are written both servers and login data.
In ~/.gnus I let Gnus scan for new mails every 5 minutes.
When Gnus starts, there comes up a program named starttls, that is
running twice, with "imap.1und1.de" and "imap.web.de".
Gnus connects correctly to the first one, but refuses connection to
web.de "access denied". This happens at startup and every 5 minutes, and
with each rescan starttls with imap.web.de is started again.
When I try to manually connect with web.de by pressing Alt+g, it works,
and again a copy of starttls comes up. But I have to do this again and
again, in order to scan my mailbox.
Changing the rescan value in ~/.gnus doesn't matter.
1. Is it possible to run two imap accounts on different servers with
Gnus?
2. If so, what might go wrong? 
Note: In Windows XP with Thunderbird it is no problem at all.
Hermann