Multiple access interference (MAI) significantly limits the performance of code-division multiple access (CDMA) systems. Existing multiuser receivers combat MAI using the knowledge of either all (conventional) or none (blind) of the interfering users. However, in practice, partial knowledge of the interference is often available at the receiver. For example, the base station of a cellular communication system typically has information about the interfering users within its cell only (about 60\% of the total interference). Therefore, conventional multiuser receivers cannot be used. We show the degradation in performance resulting from treating the unknown (out-of-cell) interference as noise or using the blind receiver. We then develop a new "partially blind receiver" for joint synchronization and detection that combines a nonlinear interference cancellation technique for the in-cell interferers with the linear blind minimum mean squares error (MMSE) technique for the out-of-cell users. The new receiver significantly outperforms the blind receiver and the receiver that ignores the out-of-cell interference. Additionally, due to the nonlinear interference cancellation technique, the proposed receiver can even outperform conventional linear multiuser receivers with the knowledge of all the users.