Mozilla Thunderbird has default limit of 5 simultaneous connections. However, some mail servers allow only a lower number of simultaneous connections. When the limit set by the server is reached, the above, or similar, error message might be encountered.

In order to resolve the issue, you will need to lower the number of connections allowed by your e-mail client.

The connection settings in Mozilla Thunderbird can be edited from Edit > Account Settings > Server Settings > Advanced > Maximum Number of Server Connections to Cache.