Yes, a movie's rating can change.
For instance, when it was initially released in 1969,
Midnight Cowboy was rated X. It was re-released in 1970 with an R rating, after United Artists, the film's distributor, petitioned the MPAA ratings board for reconsideration.
Of course, if an X-rated film were re-released today, it would have to be re-rated anyway, because that rating doesn't exist anymore.
According to the MPAA, A film can get an R-rating for the following reasons: adult themes, adult activity, hard language, intense or persistent violence, sexually-oriented nudity, drug abuse or other elements. I think
The Rocky Horror Picture Show still falls under a couple of those categories.