opponent has 6 seconds left, 11 moves are made, still has 6 seconds left?

The smallest amount of time to make a move on the server is 0.1 seconds and is  accomplished using pre-moves.  If someone is fast enough (and it helps if their opponent is a little slower), 10 moves can be played in one second at maximum.  It is usually less than this, but someone who is moving very quickly can easily squeeze 20 moves out of 10 seconds for example.

If you take a look at the game, I bet you will find this is similar to the game(s) you are referencing.  But feel free to post a game link, and we can take a look together.

Have seen this several times. Particularly noticeable when you're trying to flag someone. Recently had a game where the opponent's clock showed 0:00.1 seconds, and then they proceeded to make 20 more moves.

If you look at that game in the archive, you'll see that wasn't actually the case. It sounds like your client didn't update the clocks correctly after receiving your opponent's moves.

So another "get better internet, nothing else can be done" kinda thing?

Not at all what I'm saying. When moves are being made very quickly, it's possible that the client code can have issues updating the clocks accurately. I don't know how common it actually is, but the server keeps the official clocks, so in the game you're mentioning, if you look at the archive you'll see you opponent has more time on the clock and apparently sufficient time to make that many moves 

How is anyone to know when their opponent lost for a technical reason? People can just stop playing for any number of reasons. That being said, I have personally pointed out cases where I've won due to my opponent being stated to have abandoned the game, despite playing a move a second before. So in such a scenario it's obvious that you win for a technical reason, but a rare exception.

If you don't post a link to the game, nobody can determine what actually happened, so you're simply complaining.

In my experience, the clocks are often off by a couple of seconds in a time scramble. This especially happens while using the app.

Here are some reasons why your opponents clock didn't go down:

1. Time control. Some matches have time controls like 15 | 10 or 3 | 2. The second number would be how much time you gain each time you move. If time control is 3 | 2, each player gains 2 seconds after each move. Your opponent could have gained time by making rapid moves.

2. Premoves. Premoves allows player to make several moves in the span of seconds. Premoves take 0.1 form the clock, and you can do them up to 5 moves ahead. It's very likely that your opponent did several premoves back to back to conserve time.

3. Lag & Client. If the opponent is lagging, it takes longer for their moves to reach chess.com servers. The servers compensate by add back some time to the opponent. If your opponent is lagging, this is probably what happened. 

Happy to help. 

I looked at that game in your archive. The opponent made six moves between 7.0 and 6.0 seconds. You can see the time per move on the right hand side of the analysis. They had several 0.1 second premoves and a 0.4 and 0.2 second move. In no universe did they make 11 moves in a second. 

Also, you're playing 5 minute time control. Why are you busting out the first 10 moves in 19 seconds and then taking 16 seconds to decide on a simple retake?

None of this explains in detail how to USE premoves. It points out that you can use it but I have yet to see a clear explanation of how it’s done. If it can be done it’s an absolute joke. So you can make 6-10 moves in one second? That’s not chess. No human can make 6-10 moves sitting across from his opponent and hitting the clock after each move. If chess.com thinks because it’s on online game and different rules apply and that’s one of them…fine then we should be wise able to go to link and see in clear examples detailed of how to do it. Otherwise when one player uses it and knows how and another doesn’t the one who doesn’t and perhaps doesn’t even know about pre-moves is obviously at a disadvantage and losses the game not because he lost the game but because the one who does use the gimmick that chess.com provides is the winner not because he played a better game but understands the gimmick. So don’t give us that it’s available but show us how or have a box before the game that says this player uses pre- moves. I’ve lost many games when someone uses it.

https://support.chess.com/article/642-what-are-premoves-and-how-do-they-work

I've played 20 moves in six seconds.