Connection to server timed out & Cannot copy to Sent folder

[Jorey Bump]

I've recently switched all of my servers to Cyrus IMAP, and have 
encountered a problem that didn't appear during testing on my home network.

When using Thunderbird, it will periodically hang when accessing a 
folder, then a dialog will be displayed with this message:

  Connection to server mail.example.com timed out

The strange thing is that while it hangs on one folder, I can access 
other folders in the same account. I am also able to access other 
accounts on the same server, but the behaviour is inconsistent. It 
certainly doesn't seem to be related to DNS or any other network 
service, and it only seems to happen with Thunderbird. I see no strange 
messages in either the IMAP or debug logs. Other users have reported 
that they periodically get an error from Thunderbird saying that cannot 
save a copy to the Sent folder after sending a message.

After searching the web for a solution, I found that some people have 
had success with reducing the cached connections to 1, but this hasn't 
helped my case at all. My attempts to troubleshoot the problem by 
tailing the logs or capturing packets have been fruitless, partly 
because I can not reproduce the problem at will. I'm getting quite 
frustrated, and will appreciate any suggestions, especially if there is 
server configuration that would fix the problem.