Player Support Specialist
Based on the information you provided, this seems like a problem with your connection to our servers. Most often, these connection issues are caused by overzealous firewalls or anti-virus programs. Many firewalls are installed automatically with Windows, so you may have as many as 3 and not even know it.
We have a detailed guide to configuring your Firewalls to work properly with League of Legends. However, even when properly configured, the disabled modes or exceptions lists often do not function properly for 3rd party firewalls. In these cases it may be necessary to uninstall them. If you'd still like to give configuration a shot, you can view the article here:
However should these solutions not work, I do encourage you to log a support ticket https://support.leagueoflegends.com/home.
These type of issues can potentially require some back and forth, to speed up the process please also provide us with:
Information about your network configuration: https://support.leagueoflegends.com/entries/20132863-obtaining-a-networkinfo-log
A Netalyzr: Just go to http://netalyzr.icsi.berkeley.edu/ and run the test. Once the test is complete, copy the full url and paste it into the support ticket.
A process list: https://support.leagueoflegends.com/entries/20132988-obtaining-a-process-list