Culbertson studied aerodynamics at the University of Michigan, and continued on the research staff there after graduation. He came to Convair at Lindbergh Field in 1952, working as an aerodynamics consultant on the F-102 and F-106 interceptors for the corporate Vice President for Engineering. Culbertson returned to Michigan to complete his doctorate, then returned to Convair. He worked in Convair's Atlas ICBM program, and then was named head of the Atlas Launch Vehicle Program Office. Here he supervised conversion of the Atlas missile into a manned booster for the Mercury program, and as a deep-space launch vehicle through use of the Agena and Centaur upper stages. He retained this position until a management reorganization in 1963 made him head of Systems Engineering and Reliability for the Launch Vehicle Program. A year later, Culbertson headed the technical team for Convair's losing Manned Orbital Laboratory proposal to the Air Force.
Culbertson then left Convair and joined NASA in August 1965, being named head of the Advanced Manned Lunar Mission Studies Office at headquarters. By the end of the sixties it was clear Congress was not going to fund these post-Apollo plans, and in 1970 Culbertson was head of the Advanced Manned Missions Planning Group in the Office of Manned Space Flight. Here he worked on the Skylab and Apollo-Soyuz Test Programs. In 1979 he was named assistant for Space Transportation Systems in the office of the NASA Administrator. In November 1982 he was appointed an Associate Deputy Administrator of NASA, concentrating on planning and development of what would eventually become the International Space Station.
In December 1985 NASA Adminsitrator James Beggs was indicted for fraud. NASA's nominal number two man, William Graham, , became acting chief of NASA, but had been working at the agency for only nine days. Culbertson was named to the new Agency post of General Manager in order to provide experienced day-to-day management. He was committed to carrying out the fast pace of shuttle launches dictated by Beggs to prove that the shuttle was the inexpensive, reliable, safe, and single national method of access to space. One month later, the shuttle Challenger exploded on the way to orbit, killing the crew and putting the entire American space program in crisis.
Following the investigation of the launch, the position of General Manager was "eliminated" and Culbertson was moved to head NASA's Planning and Policy office for a year until his retirement in 1988. Many saw Culbertson as the fall guy for politicians and other NASA managers who were much more culpable for the disaster. In the 1990's he was Senior Vice President of the External Tanks Corporation, which sought but failed to find commercial backing for creating space stations from the shuttle external tanks that were jettisoned on each mission.