Well i have managed to fix mine without changing the DNS, If the problem persists I'll try it.

EXACT SAME ISSUE

come have go at riot with me here: http://na.leagueoflegends.com/board/....php?t=3060040

Hey guys, i had this issue to, and i have just completed fixing it now.
Try changing you DNS to a public one, such as the Google DNS.
Try this video, it worked for me: http://www.youtube.com/watch?v=xdQkPVL2ERg
It also seems to have increased my internet speed.

Usually when you cannot connect to the http server, somewhere on your internet provider's backbone, a fuze is out! Or, in speak that doesn't correspond to 'A Christmas Story,' one of their DNS servers is derping. It'll still let you onto the internet, but most direct queries will flunk out. Refreshing your connection, by resetting it all, or logging into your router and forcing a DNS restart or adjustment will usually fix the issue.

Thank you for responding back in the thread! I understand your concerns with receiving the same information. Unfortunately, sometimes this basic information is quite hard to discover and find in the Forum's, and I try my best to get the most common resolutions and fixes out in the open to give the highest chance that the issue may be resolved.

However, at the same time, I recognize that certain issues may require much more in-depth troubleshooting and assistance. It is because of this that I provide a link to submitting a new Support Ticket, so that if someone is still experiencing issues, we can provide far more detailed support in a one-on-one setting.

If you are still encountering these issues, please do not hesitate to submit us a new Support Ticket!

https://support.leagueoflegends.com/tickets/new

This worked for me. I'm in Brisbane, AUS, and with the weather conditions I think Telstra have had a lot of problems and haven't had time to deal with a little fuze :P. I set the DNS to google's and it worked fine straight away. Thanks.