More than likely a 36mm

 

they didn’t make a 40mm version until very recently

 

The watch worn by LBJ was the Rolex DD 36.
In terms of other watches worn at the time, they were generally far smaller in the 1960's than are seen today.
LBJ's predecessor, JFK, wore an Omega at his inauguration that was around 30 x 25 x 6mm: https://www.acollectedman.com/blogs/journal/jfks-omega-ultra-thin
I believe that Omega did a reissue of this watch a few years back.
Just as an aside, it always makes me smile when people today talk about anything below 40mm nor being a mans watch - it seems that nobody told either of these Presidents the news!

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Would have to be 36mm.

 

36mm for sure. A 36mm Oyster has the most perfect proportions IMO, especially these older references with the more svelte lugs.

To answer your other question, it's difficult to say if 36mm was THE most common gents case diameter in this period (mid-60s I suppose) as there were still a lot of 34mm and 35mm pieces being produced then, and of course 37mm and 38mm and even upwards were popular for chronos and divers . They were definitely a little larger than the 30s-50s standards, whereby 34mm was probably the "norm". I suppose you could say that 36mm was becoming the norm in the 60s if it was not already.

 

Thank you, folks, for the info. To me Johnson's 36mm Rolex looks perfectly suited to his wrist size, and he was a large man with a substantial wrist. I'm smaller than he, and I'm trying to figure out if I could wear a 36mm dress watch. I have a 6.5" or slightly larger wrist, and 36 seems small on me now.

 

36mm for sure

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And all of this time..., I thought they wore Crickets... ; )

 

As others have said, almost certainly 36mm based on date.

 

Almost certainly 36mm.

 

Just as an aside, it always makes me smile when people today talk about anything below 40mm nor being a mans watch - it seems that nobody told either of these Presidents the news!

I'm pretty sure you'd find plenty of the clothes those presidents wore to be unacceptable in your wardrobe today. Johnson's pants would have ridden almost four inches higher, for example, and he likely wore a strap connecting his socks to his shirt.

I'm 100% fine with men wearing smaller watches, but let's not pretend that fashion hasn't changed in 50 years.

 

It would have been a 36mm, pretty sure about that.