In that critical moment that the opponent hovers his/her hand over the board, about to make the final decision.
Exactly When to Sing "J'adoube" to unnerve your opponent

Anyway, the serious answer is that there is no question, because you can only adoube your pieces when it is your turn.

I think the critical moment is when you pass the age of 10 (alternatively a rating of 1600) and you realize that you are more interested in chess than in throwing your opponent off balance through some childish tricks.

This is the first time I heard a 10 year old beat Jenium. It's okay. There's always next time.
You don't seem to follow the chess news carefully enough. I even got beaten by an 8 year old once. But even at that age he had perfect manners on the board. Certainly, a sign of a serious player. Now he is an IM. (The other 8 year old kid I played in that tourney, who offered me a draw about five times in that game, never managed to get passed 1600.)

You're exactly right. I can be a little slow on the uptake sometimes when it comes to chess news. That first 8 year old IM you describe sounds like a class act. Hopefully the other one will see the light too. I bet someone named "Speedchess0" will post next in about 8 months and then delete his comment once he realizes I predicted it.
Most of the time they have you believe players only announce "J'adoube" at random (without reading anything more sinister into their intentions).
For the more serious players, though, when is the right moment/critical position to sing it for maximum opponent unnervement?
(Nobody talks about this stuff and it seems to be kept secret for some strange reason.
You ever notice that? hmmm...)
From your own OTB experience, When do you believe is exactly the right moment?
Serious replies only, please. If no one responds it's safe to assume they're hoarding this missing piece to the chess success puzzle for themselves. Imagine that.