For a reliable prediction of the NMSSM Higgs boson signatures at present and future high-energy colliders and a proper distinction of the NMSSM and MSSM Higgs sector the precise knowledge of the Higgs boson masses including higher-order corrections is indispensable. In this paper, the one-loop corrections to the neutral NMSSM Higgs boson masses and mixings are calculated in three different renormalization schemes. In addition to the $\overline{\mathrm{DR}}$ renormalization scheme, existing in the literature, two other schemes are adopted. Furthermore, the dependence on the value of the top quark mass is investigated. From this and the variation of the renormalization scale, at one-loop level, the residual theory error due to missing higher-order corrections can be estimated to be of the order of 10%.