  We assume that the Pauli exclusion principle is violated for neutrinos, and
thus, neutrinos obey at least partly the Bose-Einstein statistics. The
parameter sin^2 chi is introduced that characterizes the bosonic (symmetric)
fraction of the neutrino wave function. Consequences of the violation of the
exclusion principle for the two-neutrino double beta decays are considered.
This violation strongly changes the rates of the decays and modifies the energy
and angular distributions of the emitted electrons. Pure bosonic neutrinos are
excluded by the present data. In the case of partly bosonic (or
mixed-statistics) neutrinos the analysis of the existing data allows to put the
conservative upper bound sin^2 chi < 0.6. The sensitivity of future
measurements of the two-neutrino double beta decay to sin^2 chi is evaluated.
