We consider two interacting systems when one is treated classically while the other remains quantum. The most general form of such dynamics can be derived and has implications for the foundations of quantum theory, and to the problem of understanding gravity when spacetime is treated as fundamentally or effectively classically. If any system is treated as fundamentally classical, the dynamics necessarily results in decoherence of quantum systems, and a breakdown in predictability in classical phase space. We prove that a trade-off between the rate of decoherence and the degree of diffusion induced in the classical system is a general feature of all classical-quantum dynamics. Applying the trade-off to general relativity enables us to experimentally test the nature of spacetime. Bounds on decoherence rates arising from interferometry experiments, combined with precision acceleration measurements, squeezes the theory from both sides and can be used to rule out theories in which spacetime is described classically.