In bilevel optimization, hierarchical optimization problems are considered in which two players - the leader and the follower - act and react in a non-cooperative and sequential manner. In many real-world applications, the leader may face a follower whose reaction deviates from the one expected by the leader due to some kind of bounded rationality. In this paper, we consider one specific instance of bounded rationality, namely follower's response uncertainty due to limited observability regarding the leader's decision. This means that the follower does not know the exact decision of the leader but only knows that the decision is in some uncertainty set around this exact decision. We consider bilinear bilevel problems and exploit robust optimization to model the decision making of the follower who faces the above mentioned uncertainty. The robust counterpart of the lower level is shown to be a bilinear bilevel problem as well so that a single-level reformulation can be obtained by replacing the lower-level problem with its Karush-Kuhn-Tucker conditions. An illustrative example is presented to emphasize the importance of this modeling aspect. Further, we establish an ex-post relation to bilevel problems with uncertain right-hand side data in the lower level.