I think it depends on the body style. To my eye, 16's look ok on pickups, coupes, and maybe roadsters; on Fordors and Tudors, they just give it the look of an interrupted hot rod project, IMO.
Looks aside, there must be good reasons the whole world changed to the fatter tires, like road stability, stopping power, and safety. They do make car feel a bit more like a rod--more fun to bomb around in. However, they are noticeably harder to steer while stopped or barely rolling. Also, they are like switching from a standard (3.78) to a 4.11 rear end, at least from my 21" tires.
I had solid 16's on my car when I first got it many years ago, but I switched back as soon as I had the money. More recently, I borrowed a set of 1935 spoked 16's with tires mounted to try out again. I was planning to switch to 16's because of the glowing reports from our club members who have switched to 16's. I was annoyed to discover I still really liked the looks of the 21's. So, I invested in some pretty Firestones.
There are other options. In 1939, my dad had something else on his "A". I don't know if those are 17" or 18" tire, among his many mods.
Steve