A few minutes in, when Lieutenant Colonel Clegg Forbes picks up the newspaper he wrinkles it quite a bit. However, the next shot shows Major William Gart's hands holding a perfectly flat newspaper with no wrinkles.
In the bar retrospective at about 9 minutes in, the beer in the glasses suddenly changes from half-head to nearly full glasses with barely any head.
In Rod Serling's opening narration, we are told that the 'X-20 is an experimental Air Force interceptor that has been 900 miles into space'. That altitude is far beyond the capability of fighter jets; even if the X-20 *did* attain that high of an altitude, the aircraft and its crew would be exposed to the radiation of the Van Allen belts. By comparison, the space shuttle typically operates at altitudes between approximately 180-240 miles. To be fair, in 1959 this was not well understood yet.
When Lt. Col. Forbes and Col. Harrington depart Maj. Gart's room, they both have their US Air Force uniform covers (caps). Col. Harrington's cap has "scrambled eggs" on the visor as it should, however, Lt. Col. Forbes' cap does not. This is an error- as *all* US Air Force officers the rank of Lt. Col. and above should have the "scrambled eggs" on their cap visors. This is current today, and was the policy in 1959, as well.
Major Gart was hospitalized with a broken leg, established early in the show. Near the end he jumps out of the hospital bed and runs to the door, but nothing seems to be wrong with his leg and no cast is visible.
In the newspaper caption beneath the photo of all three astronauts, the word "Airforce" is misspelled. It should be two words: "Air Force." BTW, in the version of the newspaper that shows only Major Gart, the article's headline spells "Air Force" correctly.