Hi - I used charter and have the exact same issue with my email - works everywhere except my macbook, I turn it on, it gets mail the first time, then no more mail unless I quit and restart it. I find it hard to believe its a charter issue since it works everywhere else (iphone/ipad) i think its an apple mail issue.
Between this and the calendar reminders that dont fire on my iphone (causing me to miss meetings) im wondering why im such an apple fan boy.
I was using Apple Mail on OS X 10.6.8 and OS X 10.9 Mac's today with Charter IMAP and the only way I could get new messages to show up was to quit out of Apple Mail and start it up again.
Thunderbird worked fine and so did Postbox.
I had problems with Entourage 2008 and 2011 giving me a 1025 error "FETCH could not complete for one or more messages."