Continuity mistake: After fighting against the shark, Bond swims upwards and there's a lady holding a fishing rod. The position of her hands changes between shots.
Continuity mistake: When Bond meets Fatima he says he wants to fish. Her hair is behind her shoulders in the wide angle, but over her shoulders in the close-up. (00:47:14)
Revealing mistake: After Nigel and Bond meet and stop walking, there's a group of bystanders behind staring at the shooting of the movie.
Continuity mistake: After Fatima dies, the position of her shoes changes between the first and second shot.
Visible crew/equipment: After the motorbike jumps over the car, a close-up of the car's rim reflects the camera rig.
Continuity mistake: When Bond closes the door to stop the shark from entering the cabin, he uses both hands. A shot later the right one disappears.
Continuity mistake: Largo's men leave the fishing boat in full on wetsuits to recover the warheads but the men who retrieve them are wearing sleeveless wetsuit tops and wetsuit shorts, also the wrong wetsuits for the colder UK waters.
Continuity mistake: During the title sequence two villains are playing an implausible-looking chess game which includes two black bishops both positioned on black squares. Black takes white's queen. A moment later we see the player putting the taken queen aside but numerous pieces have moved, including two knights and several pawns. (00:03:05)
Revealing mistake: Before Bond throws the urine specimen in Lippe's face during their fight at the health farm, look at the beaker and you can see a tablet dissolving in the water.
Revealing mistake: When Fatima is blown up by the explosive charge, there is a wide shot of a dummy standing in her place.
Continuity mistake: When the XT-70s are launched, you see Polaris A1 missiles (single warhead) being launched, In the next shot you see Polaris A3 missiles (multiple warheads) flying.
Visible crew/equipment: Before the chase between Bond's bike and Fatima, right when Bond opens the box where the bike is in, a lighting screen is reflected on the front light of the bike.
Continuity mistake: When Bond's bike is trapped in the tunnel, two white and black cars stop behind him, and two baddies with chains exit. When the angle changes, Fatima exits her car and everything's repeated: the two cars come to a stop and the baddies exit again.
Continuity mistake: During the credits Bond jumps into a window. From the outside, the window to the left of where Bond jumps through, is complete; from inside the house the window is shed to pieces.
Audio problem: In the tango-scene the music doesn't match the moves. And since Bond is an excellent dancer, he wouldn't make such a mistake. (01:17:30)
Revealing mistake: When Jack's car crashes, the actor with a military hat is replaced with a stunt driver wearing a silver helmet.
Revealing mistake: After Bond throws urine at the thug, and he collapses with all the glasses stuck to his back, note the protruding pad underneath his shirt where everything's been attached to.
Other mistake: When Fatima Blush intentionally water-skis straight up the ramp to the waterfront bar where James Bond catches her, she apologizes, "How reckless of me! I made you all wet!" To which Bond replies with his classic line, "Yes, but my martini's still dry." However, Bond's beige sport jacket (which should instantly show any water stains) is not in the least bit wet.
Continuity mistake: Right before Bond grabs a bike and chases Fatima Blush, he is eating an apple which swaps from yellowish-red to fully yellow, between shots.
Continuity mistake: While riding the bike, Bond's helmet swaps from bright to matte between shots.
Chosen answer: Neither is a professional dancer. Having studied dancing some thirty years before does not mean someone has retained the same level of skills. As you mentioned, Connery was in his 50s by that time.
raywest ★