This is normally due to the configuration of the IMAP server (such as Courier-IMAP) which may be configured to only allow a certain number of connections per IP address. Fortunately you can usually solve the issue in Thunderbird by changing the number of cached connections in the advanced settings.
Call up Thunderbird,
go to your Account Settings.
Click on Server Settings
Select Advanced Tab
Select OK and I would recommend restarting Thunderbird
Change Maximum number of server connections to cache to 1