We study the thickness dependence of the resonances of split-ring resonators (SRRs) in the optical regime using experiments and numerical simulations. It is shown that the so-called LC resonance frequency of an SRR increases with its thickness. We use inductor-capacitor (LC) circuits and a plasmonic model to analyze the resonant behavior of the SRR. The plasmonic interpretation based on an equivalent cut-wire is proven to be a good model to describe the thickness dependence of the resonant behavior of the SRR in the optical regime.