The classical notion of cognitive impenetrability suggests that perceptual processing is an automatic modular system and not under conscious control. Near consensus is now emerging that this classical notion is untenable. However, as recently pointed out by Firestone and Scholl, this consensus is built on quicksand. In most studies claiming perception is cognitively penetrable, it remains unclear which actual process has been affected (perception, memory, imagery, input selection or judgment). In fact, the only available "proofs" for cognitive penetrability are proxies for perception, such as behavioral responses and neural correlates. We suggest that one can interpret cognitive penetrability in two different ways, a broad sense and a narrow sense. In the broad sense, attention and memory are not considered as "just" pre- and post-perceptual systems but as part of the mechanisms by which top-down processes influence the actual percept. Although many studies have proven top-down influences in this broader sense, it is still debatable whether cognitive penetrability remains tenable in a narrow sense. The narrow sense states that cognitive penetrability only occurs when top-down factors are flexible and cause a clear illusion from a first person perspective. So far, there is no strong evidence from a first person perspective that visual illusions can indeed be driven by high-level flexible factors. One cannot be cognitively trained to see and unsee visual illusions. We argue that this lack of convincing proof for cognitive penetrability in the narrow sense can be explained by the fact that most research focuses on foveal vision only. This type of perception may be too unambiguous for transient high-level factors to control perception. Therefore, illusions in more ambiguous perception, such as peripheral vision, can offer a unique insight into the matter. They produce a clear subjective percept based on unclear, degraded visual input: the optimal basis to study narrowly defined cognitive penetrability.