Quantum decoherence is a central concept in physics. Applications such as quantum information processing depend on understanding it; there are even fundamental theories proposed that go beyond quantum mechanics(1-3), in which the breakdown of quantum theory would appear as an 'intrinsic' decoherence, mimicking the more familiar environmental decoherence processes(4). Such applications cannot be optimized, and such theories cannot be tested, until we have a firm handle on ordinary environmental decoherence processes. Here we show that the theory for insulating electronic spin systems can make accurate and testable predictions for environmental decoherence in molecular-based quantum magnets(5). Experiments on molecular magnets have successfully demonstrated quantum-coherent phenomena(6-8) but the decoherence processes that ultimately limit such behaviour were not well constrained. For molecular magnets, theory predicts three principal contributions to environmental decoherence: from phonons, from nuclear spins and from intermolecular dipolar interactions. We use high magnetic fields on single crystals of Fe(8) molecular magnets (in which the Fe ions are surrounded by organic ligands) to suppress dipolar and nuclear-spin decoherence. In these high-field experiments, we find that the decoherence time varies strongly as a function of temperature and magnetic field. The theoretical predictions are fully verified experimentally, and there are no other visible decoherence sources. In these high fields, we obtain a maximum decoherence quality-factor of 1.49 x 10(6); our investigation suggests that the environmental decoherence time can be extended up to about 500 microseconds, with a decoherence quality factor of similar to 6 x 10(7), by optimizing the temperature, magnetic field and nuclear isotopic concentrations.