Address on Carson's letter envelope shows a zip code. Zip codes were not implemented until 1963.
The show is set in 1943. While the girls were in the bar, they were dancing to the Bing Crosby/Andrews Sisters version of "Route 66". Bobby Troupe didn't even write the song until 1946.
Max mentions hearing Charlie Parker at a jazz festival, in 1943. Parker did not record under his name until 1945. Only a handful of New York musicians and hipsters knew of Parker during WW2.
Opening sequence is set in "Lake Valley, Idaho" yet the train station is set in an orange grove, which puts it somewhere in the southern United States. Orange trees cannot handle winter freezes.