I'm using primers over 40 years old and still have and shoot some 30-06 military ammo from the 40's. Never had an issue. I could see there being issues if you had 40 year old primers in high humid areas if they weren't in sealed ammo cans with dessicant inside the cans.
In the mid-90's there was a story going around that the Clinton administration was going to mandate that future primers have a short shelf life. That was the first scramble and shortage we had for primers. Where that story started I've no idea but I was working at a gunstore at the time and people were definitely stocking up.