"On someone else's turn" is not a time.
The unit of combat time is the round, which is about six seconds. (Player’s Handbook, page 189).
The turn is not an interval of time. It's a procedure for declaring what a character spends the round doing. If during the round you want to drop concentration on a spell, you can do that, but we resolve it on your turn because you're the one doing it.
You can say, on your turn, "I step back about ten paces, take a shot with my bow, and then stop concentrating on levitate." Or you can do it in any other order, because "at any time". What you can't do is declare a sequence of actions on your turn and then, when you see what someone else is doing, say "Wait, right before she attacks, I wanted to stop concentrating." That would open the door for them to say "Then I want to attack one second earlier, before he stops concentrating."
Initiative order tells us who has to commit to a course of action first.
If you really want to do something at a specific moment relative to someone else's action, you can take a Ready action. Normally, you can only use that to take an action or to move. On the other hand, the rule about dropping concentration says "no action required", not that you can't improvise an action to do it.
However, even if you do Ready an action for "if someone attacks", they get to finish attacking before you interrupt their turn. There are reactions that will preempt the triggering condition (the shield spell, for one) but the Ready action generally doesn't.
