The electrodeposition of Cu from diluted CuSO4 in 0.5 M H2SO4 on columnar structured Pt electrodes was studied through voltammetry. For a constant set of experimental conditions, as the Pt electrode roughness is increased, bulk Cu electrodeposition tends to disappear, the amount of electro-adsorbed H atoms increases, and the coverage of Pt by 2-D Cu domains decreases. This means that under non-equilibrium conditions for diluted Cu2+-ion-containing solutions and highly rough Pt substrates, 3-D Cu nuclei, and 2-D Cu and free Pt domains are simultaneously present at the initial stages of the process. In these cases, bulk Cu can be detected when the degree of Pt surface coverage by Cu atoms exceeds 0.5.