1and1 email account slowly going offline

Are they really in an “offline” state or are they in a “failed” state (the word in parenthesis in the mailbox name)? (Mailboxes in the “failed” state are eventually taken online again.)

Do you ever get any error messages?

You can keep the Activity Viewer (⌥⌘0) open and when you notice a mailbox that goes online then try to review the connections to the server. One of them should contain 1 or more error messages which might explain why MailMate took the mailbox offline.

You're quite right, it says (failed) rather than offline.

Is there anything in this that would explain it?
Weirdly it was the Archive and Spam folders that had failed at this point

S: *
S: D21 OK UID FETCH completed
Disconnecting
Trying to disconnect nicely...
C: R22 LOGOUT
S: * BYE Server logging out
S: R22 OK LOGOUT completed
Clearing connection to imap.1and1.co.uk
Synchronizing mailbox (attempting to do it quickly): Drafts
Selecting mailbox: Drafts
Clearing connection to imap.1and1.co.uk
Trying to connect to imap.1and1.co.uk on port 993
Successful connection.
Initiating SSL/TLS...
Setup of SSL/TLS connection complete.
S: * OK IMAP server ready H mibap5 96217
C: D0 CAPABILITY
S: * CAPABILITY IMAP4rev1 CHILDREN ENABLE ID IDLE LIST-EXTENDED LIST-STATUS LITERAL+ MOVE NAMESPACE QUOTA SASL-IR SORT THREAD=ORDEREDSUBJECT UIDPLUS UNSELECT WITHIN AUTH=LOGIN AUTH=PLAIN
S: D0 OK CAPABILITY completed
C: D1 AUTHENTICATE PLAIN ••••••••••
S: D1 OK AUTHENTICATE completed
C: D2 CAPABILITY
S: * CAPABILITY IMAP4rev1 CHILDREN ENABLE ID IDLE LIST-EXTENDED LIST-STATUS LITERAL+ MOVE NAMESPACE QUOTA SASL-IR SORT THREAD=ORDEREDSUBJECT UIDPLUS UNSELECT WITHIN
S: D2 OK CAPABILITY completed
C: D3 ID ("name" "MailMate" "version" "r3905" "vendor" "Freron Software" "contact" "imap-id@freron.com")
S: D3 OK ID completed
C: D4 SELECT Drafts
S: * 4 EXISTS
S: * 0 RECENT
S: * FLAGS (\Answered \Flagged \Deleted \Seen \Draft $NotJunk NotJunk)
S: * OK [PERMANENTFLAGS (\Answered \Flagged \Draft \Deleted \Seen $NotJunk NotJunk *)] Unlimited
S: * OK [UIDNEXT 23224] Predicted next UID
S: * OK [UIDVALIDITY 706018340] UIDs valid
S: D4 OK [READ-WRITE] SELECT completed
Attempting fast synchronization.
C: D5 UID FETCH 23224:* (UID)
S: * 4 FETCH (UID 23223)
S: D5 OK UID FETCH completed
Disconnecting
Trying to disconnect nicely...
C: R6 LOGOUT
S: * BYE Server logging out
S: R6 OK LOGOUT completed
Clearing connection to imap.1and1.co.uk

Sorry for the late reply. Please try the latest test release: Hold down ⌥ when clicking “Check Now” in the Software Update preferences pane (r4025). Let me know if you still have this issue after that.