When Samantha is participating in her first debate at Oklahoma City, she mentions World War I in her argument. In 1935, this war would have been referred to as the "Great War", since a World War II had not yet occurred.
When Prof. Tolson lectures his students about the origin of the word "lynching," he refers to the Willie Lynch Letter or Willie Lynch Speech, which are generally considered modern-day hoaxes. There is no historical or archival record of either before the 1990s, when they first appeared on the Internet. No 1930s professor could have cited them.
At the Harvard debate the radio coverage is announced as WNBC. Those call letters didn't come into use until 1946. In 1935 the flagship station of the National Broadcasting System was WEAF in New York City, which headed up its Red Network. NVC's other network was the Blue Network, headed by WJZ in NYC.
At one point, Mr. Tolson says "injustice anywhere is a threat to justice everywhere." This quote was originally attributed to Dr. Martin Luther King, Jr. in his "Letter From Birmingham Jail," which was written in 1963.
When James gets home after being selected for the debate team, he is holding his books with their spines towards his left arm. But when he sets them down to go speak with his father, the spines are towards his right.
On the train, the combine is placed between the coach and the observation car, and the baggage car is in the back. At the time, large pieces of freight, like combines, were put at the front of the train, behind the baggage car.