  We address nonlocality of a class of fully inseparable three-mode Gaussian
states generated either by bilinear three-mode Hamiltonians or by a sequence of
bilinear two-mode Hamiltonians. Two different tests revealing strong
nonlocality are considered, in which the dichotomic Bell operator is
represented by displaced parity and by pseudospin operator respectively.
Three-mode states are also considered as a conditional source of two-mode non
Gaussian states, whose nonlocal properties are analyzed. We found that the non
Gaussian character of the conditional states allows violation of Bell's
inequalities (by parity and pseudospin tests) stronger than with a conventional
twin-beam state. However, the non Gaussian character is not sufficient to
reveal nonlocality thorough a dichotomized quadrature measurement strategy.
