The ordinal pattern-based complexity–entropy plane is a popular tool in nonlinear dynamics for distinguishing stochastic signals (noise) from deterministic chaos. Its performance, however, has mainly been demonstrated for time series from low-dimensional discrete or continuous dynamical systems. In order to evaluate the usefulness and power of the complexity–entropy (CE) plane approach for data representing high-dimensional chaotic dynamics, we applied this method to time series generated by the Lorenz-96 system, the generalized Hénon map, the Mackey–Glass equation, the Kuramoto–Sivashinsky equation, and to phase-randomized surrogates of these data. We find that both the high-dimensional deterministic time series and the stochastic surrogate data may be located in the same region of the complexity–entropy plane, and their representations show very similar behavior with varying lag and pattern lengths. Therefore, the classification of these data by means of their position in the CE plane can be challenging or even misleading, while surrogate data tests based on (entropy, complexity) yield significant results in most cases.