I've come across this as a developer when working for a software startup. The PM says he set a deadline for the project with the client.
And they didn't. He lied.
Why?
It turns out, as long as we believe in something to be true, it might as well be true and our bodies react to it all the same.
This applies to deadlines too.
Even when we know the date is different, the lie becomes truth and we deliver "on time".
But why do it in software?
Because the mindset of developers switches from "as many features as we can cram inside" to "ensure what is already there works and is stable", which is often exactly what the client needs.
So, as a manager, don't lie. But do say:
Our deadline is here, not later, and this is the date we're using everywhere, so we don't forget about this one. This one's important.