This problem tends to be more frequent in countries outside of North America. I'm currently trying to work with one particular ISP in the Netherlands to see exactly what this problem is, and what we can do to fix it. Right now, the ISP in the Netherlands is planning to update some of their DNS servers early next week to try to solve the problem.
Can some of you ping beta.lol.riotgames.com using a command line? The request will probably time out, but I want to see if the URL resolves to the correct IP.
i ping it and get the result is all of the trial timeout