After admonishing the newly arrived Lt. Leak on his appearance, Maj. Whittlesey says "Oh, Lieutenant; Welcome to the 308th." It can be easily seen that Whittlesey has actually said, "Welcome to the 307th." (K Company / 307th was also cut-off, but managed to find and fall-in with the 308th.)
When the plane is ready to take off for reconnaissance, the engine is started electronically. At the time of the war, engines were started by spinning the propeller blades manually, rather than by an electric ignition.
Private Krotoshinsky mentions how he "took the test", became an American and volunteered for military service. The real Abraham Krotoshinsky immigrated to the U.S. in 1912, worked as a barber until the outbreak of war, and did not become a citizen until after the war. Additionally he was one of the soldiers in Co K, 307th Infantry, not a replacement for the 308th.
At the end of the movie, the rolling text states that several soldiers were awarded the "Congressional Medal of Honor". There is no such thing. Section 3741, Title 10, United States Code (10 USC 3741) states that the correct name of the award is "The Medal of Honor".
Private Chinn is left behind at the phone outpost in the woods because the battalion did not have enough phone wire to reach the objective at the mill. However in one scene Major Whittlesey, while at the mill, attempts to contact him by phone but the line is cut. Since the battalion did not have enough wire to reach the mill, he should not have had the phone, it should have been with Chinn.
When entering the first skirmish with the Germans in the forest, Capt McMurtry is seen cocking his pistol. It is clearly visible there is no actual round loaded into the chamber. Therefore the gun was not loaded.