3" 16 gauges are not common but a few have been made over the years. Researcher has posted in time past the lengths of shells available in the early days (On Order). As I recall the 16 could be had in 2˝", 2 9/16", 2 5/8, 2 3/4", 3" & perhaps 3Ľ" (Not certain on that last one). I Have a 16ga Lefever in the 60K range & bearing the 1907 patent date which has 3" chambers in it. Likely ordered that way as I cannot really see anyone re-chambering to that length.
In the 1970s I was in a gun shop when a gentleman brought in a fired hull he had picked up in the woods of Franklin County TN, up in the mountains. It was a Rem SP hull, 16ga 3" marked for a load of 3DE-1˝oz-#7˝ shot. Obviously being a SP hull & carrying an ounce & a half of shot it was not a left over pre WWI shell. I, nor anyone else present, had heard of the introduction of a 3" 16ga. I did at the time send & inquiry to the American Rifleman. They likewise were not familiar with it, but surmised it was most likely an experimental load the factory was working with. It was apparently never put into production. In my opinion fortunate for us 16ga fans. The 16 is about the last hold-out which has not beem magnumized to death.