Model checking strategic abilities in multi-agent systems is hard, especially for agents with partial observability of the state of the system. In that case, it ranges from NP-complete to undecidable, depending on the precise syntax and the semantic variant. That, however, is the worst case complexity, and the problem might as well be easier when restricted to particular subclasses of inputs. In this paper, we look at the verification of models with "extreme" epistemic structure, and identify several special cases for which model checking is easier than in general. We also prove that, in the other cases, no gain is possible even if the agents have almost full (or almost nil) observability. To prove the latter kind of results, we develop generic techniques that may be useful also outside of this study.