The correct code is INBOX (all caps).
Too many connections, 500 errors? Here is a fix.
Thunderbird is normally the route I suggest, however the main problem with Mac Mail I've seen is with the IDLE setting, basically this leaves the connection to the server open to check for new messages as soon as they arrive rather than trying to connect every say 5 minutes or so.
NOTE: Before this fix will work, you may need to kill your hanging processes. You can simply ask Notable Web to kill all of your current processes.
This link describes the setting and how to change it in Mac Mail: http://email.about.com/od/macosxmail..._imap_idle.htm
And I'll copy the relevant parts here:
To have emails appear instantly in IMAP accounts in Mac OS X Mail: