Supersymmetric extensions of the standard model exhibiting bilinear R-parity violation can generate naturally the observed neutrino mass spectrum as well mixings. One interesting feature of these scenarios is that the lightest supersymmetric particle (LSP) is unstable, with several of its decay properties predicted in terms of neutrino mixing angles. A smoking gun of this model in colliders is the presence of displaced vertices due to LSP decays in large parts of the parameter space. In this work we focus on the simplest model of this type that comes from minimal supergravity with universal R-parity conserving soft breaking of supersymmetry (RmSUGRA). We evaluate the potentiality of the Fermilab Tevatron to probe the RmSUGRA parameters through the analysis of events possessing two displaced vertices stemming from LSP decays. We show that requiring two displaced vertices in the events leads to a reach in m_{1/2} twice the one in the usual multilepton signals in a large fraction of the parameter space.