It's an entirely plausible scene. First Canadian Army recorded 52 instances of attacks by friendly fighter-bombers between August 16th and August 18th 1944, resulting in 72 dead and 191 wounded; there were at least 53 significant incidents of friendly air attacks on US forces in Europe. The problem was particularly acute when ground forces were advancing rapidly (a situation that wouldn't have been helped if, say, an enterprising Private was making an unauthorised raid on a bank behind enemy lines).

To provide effective air support the Army and Air Force had to work closely together (not a natural state of affairs for the RAF or USAAF), maintaining situation maps of the location of friendly forces and establishing bomb lines beyond which targets could be attacked. Requests for air support would be relayed to a control centre, checked against the situation map, then missions allocated to specific units, ideally guided by a forward control post where available. Both air forces also operated armed reconnaissance missions, despatching aircraft beyond the bomb line to locate, gather intelligence on, and attack enemy forces.

There were many points at which the system could fail. If ground units had advanced faster than anticipated and were beyond the bomb line they could be mistaken for the enemy (clearly identifying targets from the air at high speed was very difficult); navigational errors by pilots negated the effectiveness of a bomb line. Ideally targets would be marked by methods including red or blue smoke; ground units were issued with yellow smoke to mark their own location, but this was not always effective as it could blend with the general smoke of the battlefield and not all pilots were fully briefed about its meaning (the worse case scenario being pilots assuming yellow smoke was a target marker).

Friendly fire incidents could result in a vicious cycle; if the bomb line was pushed further and further forward to protect friendly forces there was more chance of enemy units being encountered before it, reducing air force confidence in it. Ground units had identification panels that could be laid out to mark their location but if, concerned about air attack, units away from the front line laid out their panels then pilots could mistakenly attack friendly units in front of them. Experience of being attacked by friendly units could lead to ground units shooting at anything in the air, increasing the chances of friendly fire.

In the grand scheme of things casualties from fighter-bomber attacks were low compared to other causes, but they had a disproportionately damaging effect on morale and trust between units. The journal of the US 1st Infantry Division in July 1944 recorded:

"Wish you would tell the Air Corps we don't want them over here. Have them get out in front [and] let them take pictures [but] no strafing or bombing."

And from the Regimental History of the Grenadier Guards:

"Bitter experience had taught the Battalions that friendly planes, especially in forward areas and during swift advances, were often to be feared more than those of the enemy."

Procedures improved over the war, and the effectiveness of close air support (when correctly targeted) was generally accepted. Considerably more damaging was the use of high altitude bombing, especially by heavy bombers, against tactical targets. The inherent inaccuracy of bombing from altitude combined with lack of training for tactical support operations and insufficient planning and co-ordination meant that on five occasion, two in Italy and three in Normandy, short bombing by heavy bombers resulted in almost 2,000 friendly casualties, 435 killed and 1,445 wounded.

See:

Allied Close Air Support 1943-45, Ian Gooderson
Amicicide: The Problem Of Friendly Fire In Modern War, Lt Col C. R. Shrader
Fratricide - An Overview of Friendly Fire Incidents in the 20th Century, Wg Cdr C G 'Jeff' Jefford