While that statement makes a good sound bite there is a lot of misinformation in that statement. 1903-A3 are a relatively strong action and various pressure related issues would stop the incremental increases long before the action totally failed.
In an 1903-A3 blown primers and stuck cases would happen long before the action blew. A 700 Remington is considered to be a very strong action. In a 700 action the case head will pressure weld to the bolt face before the action fails. Same for the front half of the case. The case will be pressure welded to the chamber walls. I have seen it twice due to wrong powder selection. One was an accident and one was a new reloaders ignorance. Neither action blew. In both occurrences the barrel had to be removed to open the bolt.
In the days of the 80 gr Sierra bullet being the only choice for long range .5.56 ammo the AMU used a load call the "V-8" load that required virgin primed LC brass since the primer and the case heads were the hardest available. Primer pockets were stretched beyond use on the first firing. I have read that pressures tested at 78,000 psi. These were used for 600 and 1,000 yard competition. They were close enough to the edge that they would crack a bolt lug occasionally but the AMU would just throw in a new bolt.
SAAMI max pressures are what is consider safe in the weakest firearm commonly available firearm that round is fired in. The 45 Colt is a great example SAAMI max is 14,000 PSI and that is currently considered to unsafe for the early 1873 Blackpowder frames. Most current 45 Colt 1873 are available with a 45 ACP cylinder that is considered safe at 23,000 PSI. Fired in a 454 Casull or 460 S&W the 45 Colt large primer pocket cases start having case and primer issues at about 50,000 PSI if using LMP primers. Standard pistol primers will be the weak link and start giving issues at a lower pressure.
Switch to 454 Casull or 460 S&W brass with small primer pockets and thicker case heads the SAAMI is 65,000 PSI. So is a 45 Colt load that is 20% above SAAMI unsafe in a stronger system??? No in a stronger system at 20% above you are still only at 16,800 psi.
The firearm and cartridge combine are a closed loop system. The weakest link in the system determines what is the max safe pressure is. One example is the low serial number Springfield's. Due to heat treat issues it believed at least some if not all of the low number Springfield's are unsafe with SAAMI spec ammo. Some other old ball firearms that are in the same situation but the low number Springfield are the most common.