"Can anyone tell me if it's true that there is a law
(NYS or Federal, not sure) that if the defendants postpone 2 scheduled depositions that the plaintiff can petition the court to bar the defendants from ever deposing the plaintiff?"
No such law exists in any jurisdiction.
"It had something to do with the defense not being able to keep putting the plaintiff on notice and then keep cancelling as a means of delaying litigation or something to that effect."
If a party (either plaintiff or defendant) abuses the discovery process in order to delay a case, then the court may impose sanctions. In an appropriate case, I could see a court imposing an evidentiary sanction of the sort you mention, but it would be HIGHLY unusual and would require rather eggregious conduct.