The default number of maximum allowed IMAP connections in Mozilla Thunderbird is set to 5. Depending on the specific IMAP configuration on the server hosting your account, this may result in the following error message:
The solution is to reduce the maximum number of connections in Thunderbird.
This can be done from Edit > Account Settings > Server Settings (under the mail account for which the setting should be modified) > Advanced > Maximum Number of server connections to cache.
Change the maximum number from 5 to 1 and the issue should be resolved.