Recently reported experimental and theoretical studies of nanofluidic nonlinear devices, such as bipolar and unipolar ionic diodes, have yet to answer the question about the possibility of their further miniaturization. In this Article, we theoretically investigate the effects of size reduction, applied bias, and solution ionic strength in such devices. We compare the numerical solutions of the Poisson, Nernst-Planck (PNP), and Navier-Stokes (NS) equations with their one-dimensional, analytical approximations. We demonstrate that the contribution of electroosmosis is insignificant and find analytical approximations to PNP for bipolar and unipolar diodes that are in good agreement with numerical 3D solutions. We identify the minimal dimensions for such diodes that demonstrate ion current rectification behavior and demonstrate the importance of the edge effect in very short diodes.