Factual error: Sgt. Carter is wearing a plain belt buckle with his dress blues uniform. Since Sgt. Carter is Gunnery Sgt, he is a staff NCO so his buckle should have the USMC emblem on the buckle. (00:24:00)
Gomer and the Queen of Burlesque - S4-E21
Factual error: Gomer is a Private First Class and has one stripe. In his utility uniform Gomer doesn't have metal rank insignia on his collar. All Marines wear their rank on their utility uniforms. (00:06:00)
A Date for the Colonel's Daughter - S1-E10
Other mistake: The Colonel has his officer insignia on his shirt collar with a tie during the day while at the office. Later at home, he is in full uniform with tie and jacket, and he doesn't have his officer insignia on his shirt collar. (00:01:00 - 00:11:00)
Continuity mistake: While Gomer shows a little girl how to play jacks with sugar cubes, the number of sugar cubes go from 4 to 6 while the girl plays. (00:10:00)
Gomer and the Dragon Lady - S1-E8
Continuity mistake: Sgt. Carter and two buddies are drinking beer in mugs. The "Dragon Lady" flips a man - he hits the table and knocks over a beer bottle and a beer mug on to the floor. When Sgt. Carter and his buddies go back to sit at the table all three beer bottles and mugs with beer are back on the table. (00:06:00 - 00:07:00)
How to Succeed in Farming Without Really Trying - S3-E6
Revealing mistake: When the patroon is policing the area, raking and sweeping, they are not really raking or sweeping. They are just going through the motions but the rake and brooms are not touching the ground. (00:06:00 - 00:07:00)
Supply Sergeants Never Die - S2-E6
Continuity mistake: The Supply Sergeant Singer's top shirt button is buttoned as he approaches the counter, but as he continues around the counter it's not. (00:25:00)
Gomer, the Beautiful Dreamer - S4-E5
Continuity mistake: Sgt. Carter and Gomer arrive at Miss. Bunny's in Sgt. Carter's red car, but Sgt. Carter's red car was demolished by a wrecking ball in Season 2 Episode 13: Gomer Minds the Sergeant's Car. (00:07:47)
Friendly Freddy, the Gentleman's Tailor - S4-E30
Continuity mistake: Gomer's suit has one button but after he has the suit dry-cleaned the suit shrinks and gains a button to be a two-button suit. (00:09:36 - 00:17:43)
Continuity mistake: The Marine Captain is showing Gomer Pyle how to salute. When she salutes she has a watch on her right wrist, but when she is shown holding the salute the watch is not on her wrist.
Gomer and the Queen of Burlesque - S4-E21
Continuity mistake: Gomer is on a date - they come out of the movie theater and it's dark. Back at the base after his date Gomer runs into Sgt. Hacker, and it's daylight and the sun is shining. After their brief conversation Gomer tells Sgt. Hacker good night. (00:07:00 - 00:08:00)
Continuity mistake: The length of Sgt. Carter's career in the Marines varies from 14 years to 18 years.
Other mistake: Private Gilbert "Duke" Slater's name is not on the back of his USMC sweatshirt like the other Marines. On the back of sweatshirt is the name Estin HP not Slater G. (00:17:00)
Friendly Freddy, the Gentleman's Tailor - S4-E30
Other mistake: Gomer and Sargent Carter buy alike suits. They are together in the suits. Gomer doesn't start sneezing until he gets to Lou Ann's apartment and Sargent Carter doesn't start sneezing until Gomer mentioned the suits will make him sneeze. (00:10:33 - 00:18:36)
Gomer and the Queen of Burlesque - S4-E21
Continuity mistake: On the company road on the base, the background changes from day to day. First there are white houses and trees, then the next day there are no houses, just trees and telephone poles. (00:09:00 - 00:19:00)
Continuity mistake: Sgt. Carter's second button down on his shirt is unbuttoned. In the same sequence of Sgt. Carter talking with Gomer, the button is done up. (00:03:00 - 00:04:00)
Answer: If you saw the whole series run, you know that Gomer was a country bumpkin. He fumbled, stumbled and tumbled his way into one disaster after another. There was an episode that questioned why he was never promoted. However, a few years before Jim Nabors died, the real Marine Corps, gave him a ceremonial promotion to the rank of Lance Corporal, which Jim gladly accepted.
Unfortunately, I never saw all the episodes. Only a few episodes and a clip of the episode where he sings "The Impossible Dream" which made me wonder why he was still a PFC. Thanks so much for the info.