  We consider the phenomenological implications of the violation of the Pauli
exclusion principle for neutrinos, focusing on cosmological observables such as
the spectrum of Cosmic Microwave Background anisotropies, Baryon Acoustic
Oscillations and the primordial abundances of light elements. Neutrinos that
behave (at least partly) as bosonic particles have a modified equilibrium
distribution function that implies a different influence on the evolution of
the Universe that, in the case of massive neutrinos, can not be simply
parametrized by a change in the effective number of neutrinos. Our results show
that, despite the precision of the available cosmological data, only very weak
bounds can be obtained on neutrino statistics, disfavouring a more bosonic
behaviour at less than $2\sigma$.
