My university last year ended up blocking every port except 80 for the internet, so the only thing that worked was using a CGI client hosted on a website. Google should help you find an open one where you can change the server to connect to.
It's nothing you can fix since Mibbit connects all of its users through the same IP. Most IRC servers have a session limit that prevents too many connections from the same IP (as usually one is sufficient).
We highly suggest you use a different downloadable client (e.g. mIRC, XChat, ChatZilla) to connect or use the java IRC client provided by SWC.