Although it is true that African-American sailors served as mess stewards and stevedores during the war, this did not necessarily exclude these sailors from assignment to a gun crew during general quarters aboard ship or ashore, or manning the guns when the assigned crew were killed or wounded. In those days, all sailors, regardless of rate, received basic gunnery training in boot camp.
In the film, PT 109 is shown rescuing the remnants of a Marine patrol off Choiseul Island while under enemy fire, and when the Marines are being pulled on to the 109, the number "43" can be seen briefly on the front of the cabin briefly, while PT109 is clearly on the side. Possibly used the same boat for different scenes and missed changing the front number.
In the air raid scene at the start of the film, one of the crewmen manning an anti-aircraft gun is black. Non-white sailors in World War II were assigned almost exclusively to mess duties at sea and menial labor on shore. The services weren't desegregated until after the war.
Commander Ritchie (James Gregory) is in the galley having a cup of coffee when the ship crashes into a dock, and the coffee spills all over his uniform shirt. When he appears on deck moments later to see what happened, his shirt is dry and spotless.
When Cmdr. Ritchie spills the coffee on his shirt, he is seen at the mess table in PT 109's forward crew's quarters, but when he emerges on deck to see what has happened, he comes up through a hatch at the stern of the boat under the 20mm gun. This hatch is to the lazarette, a small workroom to the rear of the engine room. Ritchie would have had to (quite needlessly) pass through almost the entire boat below decks from bow to stern (front to rear) - through the galley, the wardroom, the dayroom, the engine room, and the lazarette - to emerge from that hatch, passing four other hatches to the deck on the way.
When Norman Fell's character (Drewitch) is in the PT engine room during the rescue of U.S. soldiers from the beach (about 50 minutes in), PT 109 is receiving Japanese shore fire on its port (left) side. But Drewitch is wounded by a projectile that pierces the boat from the starboard (right) side.
Just before PT 109 is rammed by the Japanese destroyer, Harold Marney, in the forward starboard machine gun turret, yells "Ship at two o'clock!" After the destroyer strikes the 109 and is cutting through the boat, Marney is seen (and heard) to scream the "Wilhelm Scream," a continuity error, as Marney was most probably killed on impact. In fact, a mannequin representing Marney is seen to be swept from the turret as the destroyer sweeps through the 109. Marney's horrified reaction and scream should have immediately preceded the ramming.
When Jack Kennedy is down in the boat inspecting his quarters and the land crab shows up by his feet, the big claw is on the crab's right side, when Ens. Leonard Thom appears in the scene approximately 35 seconds later with Jack Kennedy and Ens. Thom asked the size of the crew, Jack Kennedy replies one land crab. When the land crab appears, his large claw is on the left side of the crab.
In the film, PT 109 is shown rescuing the remnants of a Marine patrol off Choiseul Island while under enemy fire. This rescue actually occurred on November 2, 1943 - 3 months AFTER PT 109 had been rammed and sunk - but by PT 59, a PT which had been converted into a gunboat and was JFK's second and last command in the South Pacific.
The cylindrical smoke generators on the sterns of the recreated PT boats in the movie face the wrong way. Real PT boat's generators were installed with the smoke-emitting nozzle to port (left). The movie boat's nozzles are on the starboard (right) sides of the generators.
Twice in the film, LTJG Kennedy is asked what time it is, and he responds with "quarter to two" and "two o'clock." A Naval officer would always answer in military time to another sailor. He should have said "1345" and "1400."
When Ritchie shoots down the Japanese plane, you can clearly see the smoke generator hooked onto the middle part of the wing. There is nothing here that would generate smoke if it had actually been hit by gunfire (the plane may explode or catch fire due to the gasoline in the wing, but it wouldn't just smoke).
When Kennedy carries Pappy out of the water after swimming to the first island you can see the plank in the water that he walks on. The plank was put there so the actor could miss stepping on the coral. Later when Kennedy swims out there is no plank in the water and you can see him flinching as he walks on the coral.
When Cmdr. Richie comes to the PT 109 to inspect it, he arrives in a jeep. When Cmdr. Richie comes to the PT 109 to be taken to the new base he arrives in a jeep. It is the same shot in both scenes (you can tell from the duffel bag in both scenes).