This is joint work with Jörg Feldvoss (University of South Alabama) and a follow-up to our work on the cohomology of semi-simple Leibniz algebras. The difficulty here is the absence of a Hochschild-Serre spectral sequence for the cohomology of Leibniz algebras. Nevertheless, we manage to generalize the vanishing theorems of Dixmier-Barnes for nilpotent Lie algebras to Leibniz algebras, using the Fitting decomposition and methods of Farnsteiner. We prove for example that the cohomology of a nilpotent Leibniz algebra with values in a finite-dimensional bimodule vanishes as soon as its right invariants are zero. The nonvanishing theorems are more complicated than in Dixmier's setting for Lie algebras. We compute for example the cohomology of the trivial 1-dimensional Leibniz algebra in a finite-dimensional mbimodule and show that it is periodic in degree > 0.