This particular problem means you have asymmetric packet shaping going on.
That is, when you send packets to Riot, it takes path A,B,C. Then when Riot sends them back to you, it goes though A, D, C.
The failure mode you're experiencing occurs when D is working fine, but B is overloaded. Call your ISP; you'll probably need to convince a tech something is wrong here, so get some logs that show the packet loss.
If that doesn't work, you can probably get relief from this issue by using a VPN service to reroute your packets through a specific city, which usually results in any asymmetry in routing going away. On the down side, this typically costs you both some money ($5 a month is a common price) and some ping (typically +5-10 ms), so it's better to talk to your ISP first and see if they can resolve the problem.
edit - Forgot a basic thing. Make sure you're not playing on a wifi connection. If, for some masochistic reason, you prefer to play on wifi, make sure to run diagnostics on wifi interference and get on the least noisy channel possible.