During the US/Japan All-star Game, Berg introduces himself to Japanese dignitaries in a humorous way and (in Japanese) starts by saying he can't speak Japanese. The movie does not tell us, but this was Moe Berg's second trip to Japan. In 1932, he and a couple other American players were recruited to come teach American-style baseball to the Japanese. Berg, who had a gift for languages, taught himself passable Japanese on the ship while on the way over. Early in the cruise, one of his fellow players asked if he could speak Japanese and Berg said "No". Once they got to Japan, they were astounded to find him speaking the language. One of them said "I thought you couldn't speak Japanese?" He replied, "That was two weeks ago."
Moe Berg parachuted into Yugoslavia before his Heisenberg mission. The Yugoslavia mission was to determine which partisan group the Allies should back. Berg's report said that the group led by Tito was the strongest, and his report encouraged Allied leadership to back Tito. Tito ended up after the war becoming the dictator of Yugoslavia. None of this was shown in the movie, though it did portray him working a desk studying the Balkan region, which included Yugoslavia.
The headline for Moe Berg's obituary in Newsweek magazine in May 1972 was "3rd String Catcher; 1st String Spy."
When Samuel Goudsmit is talking to Moe Berg about Werner Heisenberg, he says that his parents were deported from Holland to the Auschwitz concentration camp. He appealed to Heisenberg for help but got no reply. He says he doesn't know whether his parents are dead or alive. The movie did not answer this question, but, unfortunately, his parents were murdered in the Auschwitz concentration camp.
In the scene where Bill Donovan (Jeff Daniels) meets Moe Berg (Paul Rudd) and shakes hands, the photo in the background is the real Bill Donovan, founder of the OSS and considered the Father of the CIA.