The Boltzmann-Nordheim equation describes the evolution of the distribution of energies of a quantum weakly interacting many particle system. It has been conjectured in the physical literature that the solutions of the Boltzmann-Nordheim equation might blow-up in a finite time. Such blow-up is expected to be related to the dynamical formation of Bose-Einstein condensates. In this talk it will describe a large class of initial data for which it is possible to prove blow-up in finite time for the corresponding solutions of the Boltzmann-Nordheim equation. (Joint work with M. Escobedo).