Revealing mistake: During the televised fight, in the wide angle shots, you can tell it's Spock's stunt double fighting.
Revealing mistake: Kirk uses a sword to cleave a Roman guard during his escape from the TV studio, but there is no blood on it. Later, though, when Claudius stabs Merrick in the back, there is blood on the knife blade.
Revealing mistake: Kirk uses a submachine-gun to shoot the lock of the cell holding Spock and McCoy, but no bullet holes appear, nor is the lock damaged.
Revealing mistake: A Roman guard empties half a magazine of bullets from his submachine-gun into Flavius, but there are no bullet holes in his body, nor blood on the floor from the wounds inflicted by the bullets.
Revealing mistake: McCoy is knelt down with the rather comically named Flavious Maximus attacking him. On the remastered version it is obvious it is a stunt double doing the part. (00:34:20)
Revealing mistake: After Kirk and party are captured by the police, a montage of scenes depicting the planet's buildings plays while the log entry is heard. One of the buildings shown, though hard to read, has the title "Massachusetts Institute of Technology" (M.I.T.) above the columns.
Answer: Kirk was getting his physical and Dr. McCoy probably turned off communications, because if he hadn't, Kirk would have left and headed straight for the bridge, leaving McCoy irritated.