The early space race seems to have been a tug and pull between the Air Force and NASA in exploration of space. The Air Force, even into the 60's, thought it could still be a player in space and tried to be with projects like the Dyna-Soar and MOL (later to be canceled before production). NASA, if memory serves, felt threatened by this, and did it's damnedest, with the support of politicians as well I believe, to keep the military out of space and to make it only NASA's territory.
The question is, could both have coexisted in space? The problem I would see is one could overshadow and drown out the other, which I think was the fear of both parties in relation to one another.