Denholm Elliott's character's name is listed as "Vognic" in the end titles but as "Vodník" in the border guard's list at the beginning of the film. While "Vodník" would be acceptable Czech name, "Vognic" is not.
Alex's facial hair when Netusil confronts him in Milena's flat: in some shots he is clean-shaven and then in others has a considerable five o'clock shadow.
The scene at the Bratislava border check point the Czechoslovak flag is upside down ---about 2 minutes 15 seconds into the film.
Stepan's file says that he is fluent in Russian, English, German and Hungarian, and can read French; but it makes no mention of either Czech or Slovak, which is not plausible. It also says he was in the "Czech Airforce" instead of the Czechoslovak Air Force.
Near the beginning of the movie, when the Czechoslovakian border guard checks the names on his list, the list contains several Czech swear words instead of personal names and occupations ("Mrdac," "Kurevnik," "Prdelac"...).
The U.S. Army Medical Corps Lieutenant Colonel, presumably a psychologist, has a full beard. This is a willfully gross violation of the appearance regulation, in or out of uniform, most especially for a commissioned officer. The film is set in 1979 (year shown in an appointment book). In that era, and for some years surrounding 1979, U.S. Army personnel were not allowed to have beards in any manner, shape or form. Sideburns could not extend below the ear opening and mustaches, which were allowed, could not extend vertically below the upper edge of the upper lip, nor horizontally beyond the opening of the mouth, making them very modest looking. The only case-by-case waiver exceptions in this era were Chaplain Branch clergy of specific religions, such as Sikh, that required their clergy to have beards as an integral part of their religious belief; a Medical Corps psychologist didn't fall under this, regardless of his religious beliefs.