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)
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)
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)
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)
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)
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)
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)
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.