Much work on the induction of non-responsiveness in lymphocytes suggests that it takes two signals to activate a cell correctly, and that receipt of the first signal alone not only causes suboptimal response, but also subsequent non-responsiveness or anergy. This paper discusses not anergy, but the original suboptimal response in the context of the well known but poorly understood phenomenon of high zone tolerance in experiments studying immunological responses to a range of antigen concentrations (low to high "zones"). We proceed to ask specific questions about the nature and timing of the signals involved. We construct three different models describing how the second signal might be received and use them to predict the shapes of dose-response curves. Comparison with data enables us to select, from a dynamical perspective, the most likely mechanism and to tentatively exclude a number of proposed mechanisms for high zone tolerance. We suggest that the stage at which a cell becomes susceptible to anergy is crucial and that it is over-contact, rather than lack of contact, which is important in the induction of tolerance.