Actually, a 'partial skeleton' is what you get when a mutilated corpse rots. Basically it means that one or more body parts was removed from the skeleton before it rotted - for example, an arm or a leg getting chopped off. It's treated the same as an intact skeleton as far as your butcher is concerned. Skeletons and partial skeletons can both be processed into bones, provided that the corpse is one which can be butchered in the first place.
In the current version of Dwarf Fortress the civilization ethics determined prohibition against butchering sentient creatures such as elves and goblins applies to their skeletons as well, which is why you can't take a elf or goblin skeleton (or partial skeleton) and turn it into a stack of bones. The same also applies to any tame animal which died for any reason other than being slaughtered at a butcher's shop, so war dogs that died in ambushes or dairy cows that die of old age can't be butchered and will leave unusable skeletons (or partial skeletons for dogs cut in half by an ambush). I have heard it claimed that skeletons may eventually rot to bones, but this seems to happen very rarely. It seems to happen more often if a creature is torn to pieces in the process of dying, such as by stepping on a weapon trap full of masterwork steel serrated disks or falling 15 Z-levels to splatter to bits. It also appears that dwarves in strange moods will sometimes grab partial skeletons to use as a source of bones.