Did not receive a response from the login servers ( Attempt 1,2,3 ) - My personal fix.
So, this problem started for me a couple days ago and I was frantically searching for a solution to it, I feel pretty silly as to how easy it was to fix for me but this may not work for everyone, I run Windows 8 so I have no idea what a Mac/Linux would be like just letting you know and I am also in the UK using the NA servers, I use BT Broadband for everyone who knows but this could work for any provider.
Anyway, here is what I did:
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I literally went over and unplugged everything (my Broadband router, tv broadband box etc) and I left it off for 60 seconds.
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I restarted my computer, let it all come back on again then run your League of Legends as ADMINISTRATOR.
This method worked for me after hours of going crazy for a solution, I do hope it works for you but if it doesn't I'm really sorry and I hope you find another its just that I messed around with Internet Explorer and all that but none of it worked for me at all. But looking through the forums this seems to be a pretty bad bug, whether its a problem on our side of the connection or not I really think Riot should make an official solution for it.
Hope it worked for you! Don't panic you will find a way!
{{summoner:10}}