Do you have a firewall or router? Open up ports 6666, 6667, and 6668 by going to the router/firewall IP. It might work. Personally I haven't had to ever do that, but you might need to. Alternatively, try connecting directly to a server like neskaya.esper.net instead of irc.esper.net. I do that for some reason, even though there's no real forseeable benefit. I dunno.

If you're using a university network, maybe the network admin has blocked all ports besides the typically used ones like Port 80, 20, 21 and such. I mean, try other IRC servers and see if you can connect from the list that comes with mIRC. If you can't connect to any of the ones you try, it's probably the case. Also, have you tried using this site's Java IRC client?

I dunno, it's really odd considering it works for me and many others. So it has to somehow be related to your network. Do you have to use proxy settings to connect to the Internet? As hard as it may be, why not try and contact the network admin and ask them if it's possible they've blocked the necessary ports and if so to maybe open them. I dunno. ;__;