We discuss the tidal interaction in neutron star-neutron star and neutron star-black hole binaries and argue that they will not be tidally locked during the gravitational inspiral. More specifically, we show that, for inspiraling neutron stars of mass greater than about 1.2 solar mass, the shortest possible tidal synchronization time exceeds the gravitational decay time, so that the neutron star cannot be tidally locked prior to tidal disruption, regardless of its internal viscosity. For smaller mass neutron stars, an implausibly large kinematic viscosity - nearly the speed of light times the stellar radius - is required for tidal locking. We also argue that the mass transfer which occurs when the neutron star reaches the tidal radius will be unstable in neutron star-black hole binaries, and the instability will destroy the neutron star in a few orbital periods. The implications of our work for the detection of these sources by LIGO and other gravitational wave observatories and for the gamma-ray burst scenarios of Paczynski (1986, 1991) are discussed.