When Joe opens a homemade card, there are two flowers inside. When shown again the flowers are missing.
The newsreel in the movie says Jesse Owens is from the University of Ohio. While there is an Ohio University in Athens, Ohio, he was actually a student at The Ohio State University --- so named in 1878, well before 1936.
The railway send-off scene to Poughkeepsie, NY, shows an English Great Northern Railway train and not an American Great Northern Railway train.
When the team is to travel from one part of the US to another via train, they board a British train with doors at each compartment, pulled by a British steam locomotive.
In the Poughkeepsie scene, one spectator boat had its name on the back on a large white banner using the Arial font. The Arial font was created in 1982 by two Microsoft employees wanting an alternative to Helvetica, 46 years after the year (1936) in which this movie was set.
Joe's father in America is shown following the race on a radio. The radio is a British-made Bush radio rather than an American model.
In the scene where coach announces he's going with the JV team, the band is using white fiberglass sousaphones. They weren't invented until the 1960's.
The railroad cars and trains in "Seattle," "Poughkeepsie," and along the "Hudson" River are all British.
Although there were a few live coast-to-coast broadcasts in the 1930s (President Franklin D. Roosevelt's "Fire-Side Chats" come to mind), the idea that a rowing competition would be covered live by radio is far-fetched. People living in Washington State could never have heard a live rowing race taking place in New York State. Even less likely would be live radio coverage of rowing competition in Berlin. Well, maybe short-wave radio, which would have been primitive in 1936. The best coverage that could be expected would be newsreels at the local movie theater a week or two after the events.
The water used for both Lake Washington and Poughkeepsie is not filmed there. Lake Washington is HUGE and has no narrow waterways as in the movie. It appears they filmed from one bank for Lake Washington scenes and the other scenes from Poughkeepsie.