Direct stimulation of the auditory nerve via a Cochlear Implant (CI) enables profoundly hearing-impaired people to perceive sounds. Many CI users find language comprehension satisfactory, but music perception is generally considered difficult. However, music contains different dimensions which might be accessible in different ways. We aimed to highlight three main dimensions of music processing in CI users which rely on different processing mechanisms: (1) musical discrimination abilities, (2) access to meaning in music, and (3) subjective music appreciation. All three dimensions were investigated in two CI user groups (post- and prelingually deafened CI users, all implanted as adults) and a matched normal hearing control group. The meaning of music was studied by using event-related potentials (with the N400 component as marker) during a music-word priming task while music appreciation was gathered by a questionnaire. The results reveal a double dissociation between the three dimensions of music processing. Despite impaired discrimination abilities of both CI user groups compared to the control group, appreciation was reduced only in postlingual CI users. While musical meaning processing was restorable in postlingual CI users, as shown by a N400 effect, data of prelingual CI users lack the N400 effect and indicate previous dysfunctional concept building.