In Sicily, both Patton and Montgomery were afforded the opportunity to pin down and destroy substantial German forces, including the entire Herman Goering Division. Instead of engaging, pinning down, and annihilating these forces as they had been ordered, they got caught up in a race for personal glory. When Alexander issued Patton a direct order to stop pushing for Messina and engage the escaping Germans to prevent their escape, the son of a ***** responded that the “orders were garbled”, put his HQ on radio silence, and kept right on going.
Both Patton and Monty bypassed these forces, allowing the Axis to evacuate over 100,000 men, some 10,000 vehicles, including fifty tanks, over 160 guns, more than 1,800 tons of ammunition and fuel, and nearly 17,000 tons of equipment from Sicily by August 17th, while they both engaged in a pissing match over Messina to grab headlines. The German forces which escaped, including the Hermann Göring Division, caused thousands of Allied Casualties.
The results of this malfeasance were almost immediately felt during the invasion of mainland Italy. The Allies were getting the snot beat out of them in the Salerno beach head. Harold Alexander ordered Monty, who had landed unopposed in Calabria in an unsuccessful diversion operation, to attack north and relieve the Allied Forces in the Beachhead on September 9th. Monty sat on his hands, gossiping with reporters, despite further orders that ‘it is of the utmost importance that you maintain pressure upon the Germans’ on September 10th and a personal visit from Major General John Harding, Alexander's Chief of Staff telling him how critical it was to move to relieve the Forces in the beachhead on September 12th. He made up some excuse about Nazi engineers setting booby traps. There were no Germans, and there were no significant booby traps. Monty didn’t get off his *** and move north until September 15th. Before the Allies broke out of the beachead the US VI Corps took 3,500 casualties, while British 10th Corps on the southern flank lost 5,500. And the Nazis Monty and Patton let escape from Sicily would inflict tens of thousands more casualties before the Italian Campaign was over.
Then there is Patton's little "no prisoners" order to the 45th Thunderbird Division on the dawn of the invasion of Sicily as part of his race with Monty.
At Comise airfield, men of the 45th machine-gunned a truck load of German prisoners as they climbed down on to the tarmac, prior to being air-lifted out.
Later the same day, 60 Italian prisoners were cut down the same way. On July 14th, near Gela, Sergeant Barry West gunned down thirty-six prisoners he was assigned to guard. At Buttera airfield, Captain Jerry Compton, lined up his 43 prisoners against a wall and executed them.
General Bradley himself ordered Sergeant West and Captain Compton to face a general court-martial for premeditated murder. West and Compton’s main defense was that they were obeying orders issued by Patton in a speech he made to his soldiers on 27th June. Several soldiers were willing to give evidence that Patton had told then to take no prisoners. One officer made a sworn statement that Patton had not only ordered the men not to take prisoners, he had explained why they shouldn’t: ‘The more prisoners we took, the more we'd have to feed, so don’t fool with taking prisoners.’
When this was reported to Bradley, he ordered that the investigation into the murder of the Axis soldiers be dropped, presumably to protect Patton from the charge of war crimes.
Patton’s ‘take no prisoners’ speech jeopardized the lives of tens of thousands of Allied P.O.W.’s - when you think that the 45th division massacred more German and Italian P.O.W.s than the SS did American G.I.'s at Malmeady, if the news had gotten out, it would have meant open season on all Allied P.O.W.s in German custody.
And that doesn't even take into account Market Garden. We all know the basics. What a lot of people don't know is that the scene in a Bridge too Far with the intelligence briefing of Lieutenant General Frederick Arthur Montague ‘Boy’ Browning really happened.
British Photo Reconaissance (under the command of Lieutenant Colonel Frederick Cotton, one of the most interesting characters to come out of WWII, but a relatively unknown hero) briefed Browning’s intelligence officer, Major Brian Urquhart, on the presence of Armor in the area around Arnhem. Urquhart, a good soldier, immediately understood the potential consequences, and briefed Browning on the issue. When Urquhart recommended that Field Marshal Montgomery and General Eisenhower be briefed, so that the problem could be vetted and addressed, Browning categorically denied the recommendation. Browning told them in no uncertain terms that, “two slides would not postpone or derail the greatest airborne operation of all time.”
When Urquhart pressed the issue, Browning said he would have ‘no part in defeatist chatter,’ and refused him permission to raise the issue with SHAEF. Major Urquhart was relieved from his position upon trumped up allegations of ‘battle fatigue’.”
As a result, a lot of good men in the American, British and Polish Airborne forces were sacrificed to Monty, Browning and Patton's vain pursuit of glory.
So no, there competition, in my opinion, caused a lot more harm than good.