Mark,
We did make the No. 16 First Aid kits during the 1930s, so your kit is most likely from that era. They first appeared in our price list in 1935, and had bandages, gauze, cotton, Drybak adhesive plaster, a bottle of spirits of ammonia, scissors, First Aid Manual, Drybak BAND-AID® Brand Adhesive Bandages and paper cups, among other supplies. Without seeing a picture of your First Aid Kit, it's hard to tell, but it's probably a 1930s First Aid Kit.
Margaret