Can't help with your problem trader, but I finding out a bit more about mine. There *is* an incompatability between Thunderbird and the Charter imap servers. The Charter imap servers do not properly implement the standards.
[much of this translated by me, and may not be technically correct, but get's the idea across]
Some email programs send a 'redundant' SELECT command on the inbox before doing the FETCH. Thunderbird does not send an extra SELECT command on the inbox if the last selected location was the inbox. The imap server is supposed to remember the last selected. Appearantly the Charter imap server does not, so when Thunderbird tried to get new messages, the server does not know where to look. Rebooting TB causes it to send again the SELECT command and it is able to see all the new message -- until somethin new new shows up.
Sounds like Charter is using long obsolete/unsupported email software. Openwave/Openwave Email MX/Unwired Planet/Openwave Messaging/Synchronoss Technologies/Synchronoss messaging/Synchronoss Email Suite/Intermail. From a Thunderbird support expert "I lost track of exactly when the Intermail product was no longer supported but according to one bug report four years ago your ISP was using version 9.0 of the software, and your log file shows they're currently using 9.0.1."
Come on Charter, Update your server to something that is fully RFC3501 compliant.
What email server are you trying to use? Are you using mobile.charter.net? If you have a legacy account, that server will not recognize your credentials. If you have a rr.com account like I do try using mail.twc.com as the email server. Port = 993, security = SSL/TLS, normal password.