Just so you know, I would say that Riot employees probably read the bug section more than any other section - they just probably REPLY there less because it's more for correlation than solving problems.
On topic however... are you absolutely certain you have no security software active? Sometimes the windows firewall will turn itself on randomly, and other times if you have other software that includes a firewall(such as some anti-virus suites), updates to the AV portion can activate the firewall.
If you're sure you don't have one, are you connecting through a router, hardware firewall or a switch? These can interfere with persistent connections...
If you don't have any of the above, try disabling you're NIC sleep mode. It seems like a silly step, but when I was doing tech support for an ISP some of the strangest problems can stem from NIC sleep mode. Google can help you with disabling this.(I'm too lazy to type it out myself, I'm sorry).
Are you running the game as an administrator? If not, you can run into very strange issues that could include awkward random issues like this one.
Are you currently living in a place that provides you with internet(such as a dormitory)? If so, it may be related to either restrictions on the network or possibly noise generated by other peoples' systems.
Are you running this from a wired computer? If not, you may be using a channel that is 'full' in your area, which could cause strange connection issues such as the one you are seeing.
As a final thought, this could very well be an issue pertaining to your ISP but not directly. It could be that one of the servers between you and Riot is refusing to pass the connection on some of the time... if so, this could appear very randomly and only occur at times that would seem strangely coincidental. Like you connect to one game, it experiences the problem. You then try to connect to a new game and since it's on a different server(since Riot doesn't just have one server) even if it's in a similar location, it takes a slightly different path through the realm of the internet to get there and thus the connection works fine.
This type of problem can't possibly be account related(unless you seriously think there's someone at Riot manually monitoring your account and ensuring you can't connect to games... and to be honest, I don't think Riot has so much money that they can afford to employ individuals for the sole purpose of driving away potentially paying customers). I don't mean to be blunt, but it's just not possible.
Trust me on this, from over 7 years of technical support experience... probably 9 out of 10 connection issues originate within the home. Even the ones that seem strangely coincidental and feel like there's 'no way' it could possibly be occuring in that way... usually are. It's not meaning you're doing anything wrong or you're stupid or anything, it's just a fact that that is where most issues occur.