During the rise to the maximum phase of solar cycle 23, several periods of extreme solar wind conditions have occurred. During such an example on May 4, 1998, the solar wind monitors observed a period of strong southward interplanetary magnetic field (IMF) accompanied by a solar wind dynamic pressure that was 30 times higher than average. During this period the Polar spacecraft crossed the magnetopause and bow shock and experienced its first solar wind encounter. This case provides a rare opportunity to study the magnetopause, low-latitude boundary layer, magnetosheath, and bow shock and to test our ability to model the dynamic behavior of these boundary regions under extreme and highly variable solar wind conditions. In this study we use the gas dynamic convected field model to predict the time-dependent magnetic field and plasma properties upstream from the magnetopause and the location of the Polar spacecraft relative to the magnetopause and bow shock during the event. To test the accuracy of the prediction, model magnetic field characteristics are compared to the fields observed along the satellite track by the magnetometer on Polar. The predicted model plasma characteristics (density, velocity, and temperature) are compared to moments derived from TIDE observations, extrapolated to account for the higher energy portion of the magnetosheath distributions. Where ambiguities occur in identifying the satellite location, plasma distribution functions from two additional ion detectors (TIMAS and HYDRA) are used to resolve the observed location of Polar relative to the boundaries. With this procedure, carried out separately for ACE and Wind and for two different magnetopause models, observed features at Polar can be traced back to drivers in the solar wind, providing a unique opportunity to assess the evolution of the solar wind and its predictability from the solar wind monitors to the magnetopause. When the Polar apogee drifts to low latitudes in the future, it will provide more and more observations in this region. Therefore what we learn from this case can be indicative of what we will see in the future in Polar operations. The high-level correlation between the predictions and in situ measurements indicates that the solar wind monitors often provide adequate and useful solar wind conditions near the Earth. The tests indicate that in this event the vacuum dipole field magnetosphere model predicts a significantly larger magnetosphere than observed. While the empirical magnetopause model predictions are more consistent with the observations, they overpredict the response of the magnetosphere to transient southward turnings of the IMF, indicating that the magnetosphere does not respond to the southward IMF turnings on small timescales. Sometimes, there are significant differences at relatively small time scales between the measurements from the two solar wind monitors. It is possible for the solar wind conditions near the Earth to be different from the measurements made by at least one of the solar wind monitors. Cautions should be taken when interpreting near-Earth observations if the observations are inconsistent at small timescales with the predictions based on one solar wind monitor.