In the present work, we perform an assessment of several property-oriented atomic basis sets in computing (hyper)polarizabilities with a focus on the vibrational contributions. Our analysis encompasses the Pol and LPol-ds basis sets of Sadlej and co-workers, the def2-SVPD and def2-TZVPD basis sets of Rappoport and Furche, and the ORP basis set of Baranowska-Łączkowska and Łączkowski. Additionally, we use the d-aug-cc-pVQZ and aug-cc-pVTZ basis sets of Dunning and co-workers to determine the reference estimates of the investigated electric properties for small- and medium-sized molecules, respectively. We combine these basis sets with ab initio post-Hartree-Fock quantum-chemistry approaches (including the coupled cluster method) to calculate electronic and nuclear relaxation (hyper)polarizabilities of carbon dioxide, formaldehyde, cis-diazene, and a medium-sized Schiff base. The primary finding of our study is that, among all studied property-oriented basis sets, only the def2-TZVPD and ORP basis sets yield nuclear relaxation (hyper)polarizabilities of small molecules with average absolute errors less than 5.5%. A similar accuracy for the nuclear relaxation (hyper)polarizabilites of the studied systems can also be reached using the aug-cc-pVDZ basis set (5.3%), although for more accurate calculations of vibrational contributions, i.e., average absolute errors less than 1%, the aug-cc-pVTZ basis set is recommended. It was also demonstrated that anharmonic contributions to first and second hyperpolarizabilities of a medium-sized Schiff base are particularly difficult to accurately predict at the correlated level using property-oriented basis sets. For instance, the value of the nuclear relaxation first hyperpolarizability computed at the MP2/def2-TZVPD level of theory is roughly 3 times larger than that determined using the aug-cc-pVTZ basis set. We link the failure of the def2-TZVPD basis set with the difficulties in predicting the first-order field-induced coordinates. On the other hand, the aug-cc-pVDZ and ORP basis sets overestimate the property in question only by roughly 30%. In this study, we also propose a low-cost composite treatment of anharmonicity that relies on the combination of two basis sets, i.e., a large-sized basis set is employed to determine lowest-order derivatives with respect to the field-induced coordinates, and a medium-sized basis set is used to compute the higher-order derivatives. The results of calculations performed at the MP2 level of theory demonstrate that this approximate scheme is very successful at predicting nuclear relaxation hyperpolarizabilities.