  When a quantum system interacts with multiple reservoirs, the environmental
effects are usually treated in an additive manner. We show that this assumption
breaks down for non-Markovian environments that have finite memory times.
Specifically, we demonstrate that quantum interferences between independent
environments can qualitatively modify the dynamics of the physical system. We
illustrate this effect with a two level system coupled to two structured
photonic reservoirs, discuss its origin using a non-equilibrium diagrammatic
technique, and show an example when the application of this interference can
result in an improved dark state preparation in a $\lambda$ system.
