MHD simulations are here applied to aid in the interpretation of three apparent cusp encounters by the Cluster 4 spacecraft in unusual places when the magnetosphere was under extreme solar wind and interplanetary magnetic field (IMF) conditions associated with the passage of magnetic clouds imbedded within fast ICMEs. At the time of each cusp encounter the IMF was very strong, generally northward in one case, generally equatorial in a second case, and generally southward in the third case. In the southward IMF case, the MHD models locate the origin of the cusp-like plasma by showing that the position of the spacecraft at the time of encounter was engulfed in a tongue of high-pressure plasma extending from the magnetopause into the magnetosphere. This tongue points to the northern-hemisphere cusp as the source of the feature. In the equatorial IMF case an elevated-pressure feature that apparently marked a cusp encounter in the computations coincided, however, with a passage in the solar wind of a dynamic pressure pulse, thus giving an alternative interpretation of the feature. However, Cluster data unambiguously identified the event as an encounter with magnetosheath-like plasma. Given that the Cluster observations classify the event as a true encounter with a cusp-like plasma feature (and not a compression event), the model simulations can be interpreted as identifying the origin of the feature to have been the northern-hemisphere cusp even though-and this is the interesting point-the observation point was in the southern hemisphere. In the northward IMF case, neither cusp (defined as a magnetic funnel linking the magnetopause to the Earth) was directly connected to the observation point. Instead, this encounter of magnetosheath-like plasma appears to be an instance of boundary-layer formation by means of the Song-Russell mechanism in which two-point magnetic reconnection entrains magnetosheath plasma on closed field lines when the IMF is northward.