Continuity mistake: When the astronauts are putting on their space suits, Glenn gives a speech about morality. While Shepard rebuts Glenn's speech the space suit on the left of the clothes rack changes position throughout the sequence. It is at time at the right side of the section of the rack, turned toward the camera, turned away from the camera, moved close to the middle of the section of the rack. There is also an empty hanger that appears later in the shot when it was not there beforehand.
Continuity mistake: After Gus Grissom and his wife discover their refrigerator has been filled and they begin fighting, reporters show up outside their balcony. In Gus's preparation to meet with the press, he buttons up his blazer, then turns around to face the cameras. The next camera shot shows from outside the apartment and Gus buttons his blazer once again. He had already buttoned it up once and did not have time to unbutton it.
Continuity mistake: When Gordon is in the bathroom stall preparing to produce a sperm sample the camera pans down to the bottom of the stall. We can see the legs of the people in two of the stalls, Gordon and one other candidate. After Gordon says "Ok Glenn, I know that's you", the scene jumps a little. The belt of the astronaut to the left moves where the shots were joined. This was obviously two different shots that were spliced together at this point.
Continuity mistake: When John Glenn's Atlas rocket is shown both before liftoff and as it is lifting off, a hodgepodge of stock rocket footage is pieced together. Some Atlas rockets are white, some unpainted. A camera attached to the outside of the rocket looking backward shows fins at the base of the rocket. The Atlas rocket had no fins. It also shows the rocket oscillating wildly on its axis as it ascended, something no normal manned rocket would do.