I think in the case of Rogers and Hart (or any duo where one writes the tune the other the words - like Elton John and Bernie Taupin) it is still listed as a joint composition, with no differentiation about who did what.
After all many songs by bands list all the members of the band as writers, which may be 4 or 5 people, and it would be impossible to say who wrote the tune and who wrote the words as they probably all did a bit of each.
I guess the main situation where this arises is if a tune is written as an instrumental FIRST, then months or years later lyrics are added.
Then you have a situation where there is a writer of the instrumental tune on its own, and the writers of the version with the tune and lyrics combined.
That could certainly get interesting regards who gets paid what.