My problem is very similar (but worse).
I was playing an ARAB, like have been for the last few days when suddenly for the first time ever I had a lag spike that didn't produce a reconnect option in-game. It's hard for me to remember the details, b/c I was finishing dying as the spike occurred and my screen was going to grey. I remember seeing my teammates jumping around and then I waited for the in-game "reconnect" option to show up. I watched my re-spawn timer count down and then I didn't resurrect. People had stopped moving I think. I alt-tabbed and my internet was working fine for web browsing/downloading. I either exited the game or it finally crashed out (I don't remember). This produced a reconnect button in the launcher. So I tried that... and tried it again...
Each time It would produce the "League of Legends" Logo in the center screen (as usual) but then not produce the character loading screen, but rather an error saying:
Title: "Failed to Connect"
Text: "Unable to connect to the server. If you have a firewall, it may be blocking the connection. Please refer to you firewall's documentation. Would you like to retry the connection"
Options: "Retry", "Cancel"
From that moment since, I have been unable to get to the character loading screen (instead receiving the above error).
The only exception I found so far is if I watch a featured game (which works fine)
EDIT: My problem seems to be specific to my campus (I'll check with my friends who play this game later to see if they have the same problem).
As a Computer Science major in his 4th year this makes absolutely no since to me, but I found the problem to be related to my wired connection.
If I connect to my colleges wireless router, as opposed to the wired connection then games load properly. This isn't a preferred fix, since wireless is generally slower/less reliable.
Vent: Who changes router configuration over the weekend in the middle of the day? and why wasn't there any notification of this?
Conclusion: It seems something suddenly changed on the wired connection between my PC and it where it joins with the wireless connection before getting sent over fiber off into the internet.