We present formulae for the coherent and incoherent tune shifts due to the nonlinear resistive wall wake field for a single beam traveling between two parallel plates. In particular, we demonstrate that the nonlinear terms of the resistive-wall wake field become important if the gap between the plates is comparable to the transverse rms beam size. We also compare the theoretically predicted tune shift as a function of gap size with measurements for an LHC prototype graphite collimator in the CERN SPS and with simulations