Indeed, 2 9/16" 16ga shells WERE made. So were 2 5/8" 12's and 2 1/2" 20's--respectively the previous standard for all 3 gauges, before the switch was made to 2 3/4". In my 1940 Shooters Bible, which lists ammo offerings from all the American companies, I find a lot more 2 9/16" 16's than I do 2 3/4".
Not likely a NID, even with short chambers, would take much of a beating from current ammo. For one thing, most shells are slightly shorter than 2 3/4". So in an honest 2 9/16" chamber, you're likely to be shooting a shell that's not more than 1/8" too long. A strong gun like the NID probably wouldn't notice the difference, although you might feel some recoil reduction if you lengthened the chamber and forcing cone, or just the forcing cone.
My 16ga NID, made in 1936 by the serial number, has 2 3/4" chambers, and they do not appear to have been altered. I don't concern myself with what I shoot in that gun.
