Shot1 is right - the ones converted to .308 were intended to fire the lower pressure CETME round, not the .308 Win or NATO rounds. The model 1916 is actually a M95 Mauser action and isn't up to handling the pressures that the M98 can.
They should feed 7mm ball ammo, or any ammo with similar bullet profiles, OAL, etc, as that is what they were designed to work with. The .308 is slightly shorter so maybe they're a little more touchy about feeding them?
My first thought would be that maybe the magazine follower springs have weakened with age and are not lifting the cartridges up enough for a proper positioning to the bolt and chamber to feed smoothly. Just a thought.