In the WWI segment, when Truman and his men have called for a successful artillery strike against an enemy position, the men are cheering and we can clearly hear one of them shouting "Alright!"
It was not until the 1970s that people began using the word "alright' by itself as a cheer or as an exclamation of satisfaction.
In 1944, General Marshall's uniform displays both the World War II Victory Medal and a combat star to the Combat Infantryman Badge (denoting two awards of the decoration). Neither of these decorations were introduced until after the end of the Second World War. General Marshall himself was also never a recipient of the Combat Infantryman Badge, which was only awarded for front line combat duty.
Where General Marshall and President Truman are talking on the lawn in Potsdam, part of Marshall's lapel brass includes crossed rifles, which indicates the wearer belongs to the infantry. General officers do not have a branch of service or wear lapel brass indicating such. The Chief of Staff, however, does wear a star-shaped device on his lapels indicating his office. The Marshall character should have been wearing these.
Just after Tom Pendergast's death, Truman sits down at a card table to play poker. The dealer says they are playing five-card stud, but then proceeds to deal each player two cards face down. A five-card stud hand consists of a single hole card and four cards that are dealt face up.
Incorrectly regarded as goof: "Harry Truman was left-handed but the film shows Gary Sinese writing right-handed." Truman was born naturally left-handed, but he was raised to be right-handed. Truman in fact wrote right-handed, and there are many photos of Truman showing him at his desk with a pen in his right hand.
When President Truman visits the White House kitchen staff to bring them some birthday cake, he switches on a radio to let them hear news of the end of WWII. The radio comes on the instant the switch is turned- which is incorrect for a radio of that era. Radios in 1945 utilized vacuum tubes. Radios (and later televisions) that used these tubes always took several minutes to "warm up" after being switched on.
After FDR's death, there are scenes of the funeral procession in Washington, DC., followed by his funeral train leaving Warm Springs, GA. He would have left Warm Springs first, the place of his death, then proceeded to the Capitol.