Using quasi-static equilibrium theory and three-dimensional MHD simulations, we confirm earlier two-dimensional results that finite boundary deformations of magnetotail equilibria can lead to large local current density enhancements, that is, the formation of thin current sheets. Equilibrium configurations that satisfy flux, entropy, and topology conservation cease to exist when the boundary deformations exceed critical limits. This provides a strong argument for onset of instability or the loss of equilibrium, regardless of the dissipation mechanism. Catastrophe points can be reached for relatively modest perturbations of the boundary. The most important internal property that influences the critical limit is the variation of the lobe magnetic field strength with distance from the Earth, related to the overall tail flaring. Stronger flaring, corresponding to a more rapid decrease of the lobe field with distance downtail, tends to stabilize the tail. That means such configurations require stronger perturbations to reach the critical limit or show less significant thinning and current density intensification for a given perturbation. Due to the nonlinearity of the tail response to the perturbations, the current intensification tends to be more localized than the perturbation.