Recent experiments suggest that level-k behavior is often driven by subjects' beliefs, rather than their binding cognitive bounds. But the extent to which this is true in general is not completely understood, mainly because disentangling 'cognitive' and 'behavioral' levels is challenging experimentally and theoretically. In this paper we provide a simple experimental design strategy (the 'tutorial method') to disentangle the two concepts purely based on subjects' choices. We also provide a 'replacement method' to assess whether the increased sophistication observed when stakes are higher is due to an increase in subjects' own understanding or their beliefs over others' increased incentives to reason. We find evidence that, in some of our treatments, the cognitive bound is indeed binding for a large fraction of subjects. Furthermore, a significant fraction of subjects do take into account others' incentives to reason. Our findings also suggest that in general, level-k behavior should not be taken as driven either by cognitive limits alone or beliefs alone. Rather, there is an interaction between own cognitive bound and reasoning about the opponent's reasoning process. From a methodological viewpoint, the tutorial and replacement methods have broader applicability, and can be used to study the beliefs-cognition dichotomy and higher order beliefs effects in non level-k settings as well.