Might be something here that will help...
https://support.chess.com/article/213-how-do-i-fix-my-disconnect-lag-issues
Oh and don't be shy about refreshing the page if your opponent seems to be taking too long. Could be a temporary disconnect and you didn't get his last move. Good luck!
I've just started playing on chess.com, and both of my losses so far have been from "game abandonment". Example: https://www.chess.com/analysis/game/live/6009957402
Usually when this happens, I'll make a move, and then the opponent's timer will count down like normal. After some time (~ 1 minute), the game immediately shows that I have lost due to abandoning the game. I don't get any warning that I need to reconnect - no flashing timers, no notification - the game just ends. I don't have any opportunity to make a move as I never see my opponent's move.
I'm using a wifi connection that consistently has 40Mbps download/4Mbps upload. chess.com tells me that I have an excellent connection with low latency. I can play online games (WoW, Diablo 3) and while there are sometimes lag spikes lasting a second, they certainly don't last the 55 seconds that it takes for the abandonment timer to run down.
I could switch to a wired connection, but that'll be annoying and take a few days. (I'm pretty sure this will resolve the issue?) Is there anything else I can do in the meantime that may fix this?