For an everywhere defined closed linear relation in a Banach space the concept of n-invertibility is introduced and studied. It is shown that many of the results of S.R. Caradus and other authors for operators remain valid in the context of multivalued linear operators. In particular, we gather some results and characterizations of n-invertibility and semi-Fredholm linear relations. Some stability results under perturbations by compact relations are also given for this concept. Part of the results proved in this paper improve and generalize some results known for pseudo-generalized invertible operators [Filomat 36:8 (2022), 2551–2572].