Disconnected from the Live server does not mean you are disconnected from the internet.
But you can look through the following and there might be some things to help out:
https://support.chess.com/customer/en/portal/articles/1444918-my-internet-connection-is-fine---why-am-i-getting-disconnects-
https://www.chess.com/blog/News/how-to-adjust-your-live-chess-connection
Just a few minutes ago playing a game I got a message that I had disconnected. So I checked quickly in another tab where I was connected to fide and my internet connection was just fine as I could look up peoples ratings and get a very fast response. Then I was told that I had lost on time. I had 2 minutes left when I got the disconnected message. And no I wasn't using the old version of chess.com but the new one in chrome on windows. Did anyone else get disconnected like this tonight? Any suggestions about where the problem might be? I had just lost a bishop sadly to the advancing pawns of my opponent but I doubt that is any way relevant.