We consider Bianchi type I cosmologies with unidirectional magnetic and electric fields, assuming as well the existence of a global spinor field ψ(t) as one more possible source of gravity able to suppress the inevitable anisotropy accompanying a nonzero vector field. The field ψ(t) is assumed to contain a nonlinearity in the form sn, where s = ψ̄ψ and n = const (the special case n = 1 corresponds to a Dirac massive field). The structure of the stress-energy tensor of the spinor field is shown to be the same as that of a perfect fluid with the equation of state p = wρ where w = n-1. The Dirac massive spinor field and nonlinear fields with n < 4/3 are shown to be able to provide isotropization. A numerical estimate shows that this isotropization could occur early enough to be compatible with observations. © Springer Science+Business Media, LLC 2009.