Revealing mistake: If you look at the river under the bridge and all other shoots it isn't flowing yet it is quite shallow. In one shot the wind is blowing the surface.The river was purpose built along with the town on former British Aerospace Factory in Hatfield Herts and the river is a shallow oblong mini lake the runs the width of the set.
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: 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.
Continuity mistake: When James Ryan from Minnesota is told his brothers died, the arm behind him disappears and reappears depending on the angle.
Other mistake: In the final battle scene, soldiers jump on top of a German tank and throw a few grenades in. While the sound of the grenade's fragments bouncing around the tank can be heard, there is no sign of the actual explosion.
Answer: These were barrage balloons, commonly used during the war. They are used to stop low level bombing and low level fly bys by enemy fighter planes. The cables attached to the balloons are designed to cut through the wings of the aircraft and to bring them down, so any pilot would have to fly above them, and the balloons would also restrict the view from above.