I seriously doubt it, however, I'm willing to be educated.
Metal fatigues when it is constantly being worked (flexing). There's not a whole lot of flexing in a primer's life: the cup is punched out of a sheet, the finished primer gets shoved into a primer pocket, and then it gets fired.
I suspect they're referring to embrittlement of the brass primer cup due to mercury fulminate priming compound. That particular problem dates back to the 1930's, along with nickle plating of primer cups to prevent it. That is not an issue with primers using lead styphnate (basically everything since the 1960s), which is why you find plain brass (unplated) primers made by Remington, Wolf, etc.
The "lead free" priming compounds do have a more limited shelf life, but that has nothing to do with cracking or increasing pressures.
I have limited experience with using old primers, because they get loaded and shot up fairly quickly, but I know I've loaded some that are 25 years old with no issues.