Continuity mistake: When Mike crosses the bridge at the end he gets shot and is helped by a soldier. From the German view the bridge is clear, then it cuts to the American side and Mike and the soldier are exiting the bridge to their left towards Miller. A motorbike and sidecar have now appeared behind them and a crouching American soldier is next to the motorbike halfway across the bridge.
Continuity mistake: Right before Capazzo dies he holds up the letter, but it's in a zip-lock bag. After the sniper is killed the medic retrieves the soaking wet, unbagged letter.
Suggested correction: The letter is never in any bag, plastic or otherwise. It's always bare or else he wouldn't be worried if there is blood on it. Zip-lock bags weren't invented until the 1950's anyway.
Continuity mistake: When James Ryan from Minnesota is told his brothers died, the arm behind him disappears and reappears depending on the angle.
Continuity mistake: While they are preparing to defend the bridge towards the end, Mellish tells Upham to be "Johnny on the spot with the ammo." and proceeds to put 2 30 cal. belts around Upham's neck. In the first shot the bullets are pointed down, when the shot changes the bullets are pointed up.
Continuity mistake: When the unit finds the first Private Ryan, after he hears his brothers are dead, he leans towards his superior officer as he cries, but in the next scene he has his face buried in the officer's chest.