I have had this happen to me and sent a request to riot with the screenshot. This was their response:
First thing's first, I definitely wanna recommend switching from a private DNS to Google's public DNS. This will let League choose the smoothest and most efficient connection route possible.
After you've changed your DNS, don't forget to flush it afterwards! Restart your computer after you're done to finalize the changes.
And since it's super quick and easy, let's also go through a power cycle of your router and modem. Pull out the power cable on both devices (or one if you're using a router/modem combo device), wait at LEAST 30 seconds, and then plug them back in.
Once you're done with these steps, before you queue up for any more matches, let's take a closer look under the hood to see what else could be going on by using the Hextech Repair Tool:
Download and install the HRT here.
If you're having trouble launching the repair tool, check out our quick 2 minute walkthrough video here that'll cover all the bases.
The default settings will be perfect as-is, and you can hit Start right away.
Once it's done, you'll see a LoL Logs.zip file pop up on your desktop, and you can attach that to your next response just like an e-mail attachment. Thanks!
Here is the link to the video and with the HRT download in the description https://www.youtube.com/watch?v=NLQlatrS4Gc
Hope this helps, haven't tried this out personally as I don't have access to my router