Why would Anti-Virus software interfere with your internet connection? Unless it's hogging bandwidth by checking for updates and such?
Kaspersky Internet Security Problems with Live Chess

I don't use KIS but I think that problem is due to it's firewall settings.Perhaps, you just need to allow all the inbound and outbound connections related to chess.com in the KIS firewall settings.
Hope it helps.
Ah yes, it's not just Anti-Virus, it's an entire Internet Security suite. Of course . The Firewall would explain it.
How would I go about allowing these connections in the firewall settings? These things have gotten so complex over the years that it seems entirely impossible to set it up without help and a BA degree in interwebz...
I just figured it out. It is the new ads that Chess.com is displaying on Live Chess. Everytime there is a new ad, KIS intercepts it which causes a disconnection. Quite bad. Either I have to disable the anti-virus protection (which is bad because it also disconnects the firewall) or I have to have the game constantly disconnected. Oh well, such is life.
Just a heads up for anyone who is using Kaspersky Internet Security (KIS) or is thinking about using it. Ever since I installed KIS 2013 I keep getting disconnected while playing live chess. The only way to stop the persistent disconnections (it will happen several times during a brief game) is to pause KIS security. Pretty bad situation but there doesn't appear to be anyway to stop KIS from what it is doing.