I have an account with fastmail.fm. In the light of the recent SSL 3.0 insecurity discovered by Google, they've disabled SSL 3.0. Claws-mail is now unable to connect. Fortunately there's a workaround at the moment.
Until the new Claws release is out, (version 3.11.0), you can either use STARTTLS (use port 143 on IMAP, 110 on POP3, 25 or 587 on SMTP for that). Or you can set the preferences: gnutls_set_priority=1 gnutls_priority=NORMAL:-VERS-SSL3.0 in ~/.claws-mail/accountrc
Fixed in version 3.11.0.
*** Bug 3408 has been marked as a duplicate of this bug. ***