When a fluid is pumped into a cavity in a confined elastic layer, at a critical pressure, destabilizing fingers of fluid invade the elastic solid along its meniscus (Saintyves, Dauchot, and Bouchaud, 2013). These fingers occur without fracture or loss of adhesion and are reversible, disappearing when the pressure is decreased. We develop an asymptotic theory of pressurized highly elastic layers trapped between rigid bodies to explain these observations, with predictions for the critical fluid pressure for fingering, and the finger wavelength. We also show that the theory links this fluid-driven fingering with a similar transition driven instead by transverse stretching of the elastic layer. We further verify these predictions by using finite-element simulations on the two systems which show that, in both cases, the fingering transition is first-order (sudden) and hence has a region of bistability. Our predictions are in good agreement with recent observations of this elastic analog of the classical Saffman-Taylor interfacial instability in hydrodynamics.