In computer simulations of chaotic systems, small errors grow exponentially, thereby questioning the validity of the numerical results. Fortunately, shadowing techniques often show that the simulated trajectories are close approximations to the true trajectories. However, chaotic attractors containing unstable periodic orbits with different numbers of local unstable directions are necessarily unshadowable. Here we argue that this route to unshadowability should be common in coupled dynamical systems. Our argument is robust in the sense that it does not require any embedded invariant manifold or any exact condition like unidirectional coupling. Thus, it should hold for real physical systems. Numerical simulations of the forced, damped double pendulum are presented as an example.