The equations of nematic Liquid crystal hydrostatics are used to determine the driving forces that cause the breakup of catenoidal shaped liquid crystalline networks during the phase separation of isotropic and nematic phases. The catenoidal shaped liquid crystalline network is assumed to be an elastic network embedded in an isotropic matrix. The elasticity of the network arises from isotropic surface contributions (interfacial tension) and bulk orientation gradients (Frank elasticity). For liquid crystalline networks with an orientation structure affine to the catenoidal shape, the theory predicts that under certain parametric conditions capillary instabilities will break-up the network by setting up a viscous flow from the thinner sections of the network toward the thicker sections. The stability properties of the liquid crystalline network are summarized in a two-dimensional phase stability diagram, given by the ratio of surface to bulk elasticity as a function of the curvature of the catenoid. Typical parametric conditions for nematic polymers indicate that the liquid crystalline networks are unstable, in qualitative agreement with the findings of Nakai et al. (Nakai, A; Wang, W.; Hashimoto, T. Macromolecules 1996, 29, 5288).