When the space agency has the press conference saying Mary will be the first American in space, the U.S. flag in the background is wrong. There is no possible way that the stars are in that spot, unless of course somebody put the flag on backwards. (This is coming from a CANADIAN!!! you Americans should have noticed that right away!?!)
Hey, I haven't seen the movie in a while, so if I screw up, don't beat me down too bad. In the fifties or sixties or whenever this was taking place, were Hawaii and Alaska even admitted into the Union? If not, that would definately mess the stars up.
It is possible that the flag was put up in a fashion other than is normally seen. The US military, as well as all government agencies, have a specific set of rules for how the flag must be displayed depending on the circumstances. It is possible that the flag may have been required to face another way.
OK, that part in the movie was in 1958. Alaska became a state, the 49th, in 1959. Hawaii, the 50th state, became a state in (I believe) 1959, also. So clearly, the flag would be missing two stars. Perhaps this is what was going on. And by the way, just because you were the first person to post it doesn't mean you were the first to notice it.