Gravitational radiation from known astrophysical sources is conventionally treated classically. This treatment corresponds, implicitly, to the hypothesis that a particular class of quantum-mechanical states — the so-called coherent states — adequately describe the gravitational radiation field. We propose practicable, quantitative tests of that hypothesis using resonant bar detectors monitored in coincidence with LIGO-style interferometers. Our tests readily distinguish fields that contain significant thermal components or squeezing. We identify concrete circumstances in which the classical (i.e. coherent state) hypothesis is likely to fail. Such failures are of fundamental interest, in that addressing them requires us to treat the gravitational field quantum-mechanically, and they open a new window into the dynamics of gravitational wave sources.