Nearly every time I've held out and not given a number I've gotten a much higher salary than the number I would have given.
It is true, of course, that when they give you the number that they leave money on the table for negotiation, that is standard practice. It's even factored into the offer amount at bigcorps, and chances are they have an amount they cannot go over and it's set. So you should always ask for more after they lay the money out.
My experience has been about 50/50 with that. Twice when I haven't given the number first, they've pleasantly surprised me with significantly higher offers than I would have expected. Twice when I did give a number, they said "oh no that's way too low, we want you to be extremely happy with comp" and bumped me up. In all other circumstances I don't think it mattered, and I suspect that giving the number or not was moot in all 4 of those cases.
In the second example, it's likely that you still left money on the table. They almost certainly gave you the lower/midrange of your projected band which was likely significantly above the number you gave.
Hard to say from a blurb whether the situation is an outlier, just generalizing here based on on basically every salary discussion I see on the hiring side.
It is true, of course, that when they give you the number that they leave money on the table for negotiation, that is standard practice. It's even factored into the offer amount at bigcorps, and chances are they have an amount they cannot go over and it's set. So you should always ask for more after they lay the money out.