Your opponent is probably premium. What is likely happening is that your opponent is getting timeout protection, a premium feature. When that happens all their games, except ones in no-vacation tourneys, get paused.
In that process a minimum of 24 hours of vacation time get used up. If the player comes off vacation before the 24 hours have elapsed, the difference gets added to the time of the other games where it is their move.
If I am not a paying member but am playing online chess here, who can I go to for help? I've got one game where the opponent won't move any more and even though the timer for moves is 3 days, he hasn't moved in many more than that and the clock doesn't run out but adds more hours when it gets down to the teens.....How can more hours get added when he hasn't moved? I told him in chat to move or resign but no answer. How do I get rid of this game with offering a draw or reseigning?