We report on a new analysis of instabilities in close neutron star binaries based upon (3+1) dimensional general relativistic numerical hydrodynamics calculations. When a realistic equation of state is employed, orbit calculations for two 1.45{ital M}{sub {circle_dot}} neutron stars reveal surprising evidence that general relativistic effects may cause otherwise stable stars to individually collapse prior to merging. Also, the strong fields cause the last stable orbit to occur at a larger separation distance and lower frequency than post-Newtonian estimates. {copyright} {ital 1995} {ital The} {ital American} {ital Physical} {ital Society}.