It must have been machine gun brass. I've never had this problem with any other reloading. I've reloaded thousands of pistol and rifle calibers. I measured and trimmed to proper length, ran through two different sizing dies. I've never had this problem with commercial brass.
Would anneling the cases, then sizing them help?
It's time to take some measurements. Get the SAAMI diagram for .308, and you calipers, and start measuring everything you can, especially on the rounds that won't chamber. You will likely find some dimension that, even after resizing, is larger than SAAMI max cartridge. You will likely have to adjust your sizing setup, or may even have a faulty sizer but that seems unlikely.
If you convince yourself that all dimensions are correct, then you have to contemplate the possibility that some dimension of the chamber is smaller than SAAMI min chamber (which will hold the max cartridge of course). That requires a chamber cast to investigate.
