During 1996, the magnetometer and particle instruments on NASA's Polar spacecraft detected several apparent magnetopause crossings. In constast, with a geocentric apogee of only 9 RE over the northern pole, this spacecraft was not expected to leave the magnetosphere. We have investigated the modifications that are required in existing magnetospheric magnetic field models in order to account for these observations. In order to match the observed particle distributions (reported elsewhere,) this model includes northward-IMF reconnection. For the event of May 29, 1996, unusual magnetic and particle observations began at an altitude of 5 RE while the spacecraft was sunward of the expected northern cusp position. Using a cylindrically symmetric magnetopause, we are able to reproduce the measured field reasonably well but with problems in the timing of the boundary crossings. The crossing times are much improved by making an adjustment in the spacecraft position which could arise either from motion of the boundary or an indentation in the magnetopause at the cusp. The results from this model are compared against results from the analytic open-magnetosphere Toffoletto and Hill [1993> model and the empirical Tsyganenko [1996> model; significant improvement over both prior models is shown. This improvement is shown not only with respect to data from Polar but also with respect to data from the Interball-Tail spacecraft. The model indicates that the Polar spacecraft encountered regions of overdraped lobe field lines that connect to the southern polar cap but exit the magnetosphere through the northern cusp. ¿ 1998 American Geophysical Union