In this paper, we present a successive interference cancellation (SIC) scheme for a multicarrier (MC) direct-sequence code-division multiple-access (CDMA) system, using band-limited spreading waveforms to prevent self-interference. In every subband, the SIC receiver successively detects the interferers' signals and substracts them from the user-of-interest. A comparison is made among SIC, a minimum mean-squared error (MMSE) receiver, and matched filter (MF) detection with maximal-ratio combining (MRC). We also consider suboptimal combining using pilot symbol-assisted modulation (PSAM) to make the system more realistic. Analytic bit-error probabilities for SIC, MMSE, and MF in correlated Rayleigh fading channels are derived. The theoretical results for SIC, MMSE, and MF are shown to agree well with simulations. In particular, SIC and MMSE are shown to achieve better performance than MF with MRC; when the number of users is small, SIC provides better performance than does MMSE. Further, the correlation among different subcarriers is studied, and only large subcarrier correlation coefficients result in an obvious worsening of performance. Finally, we derive results for the performance degradation that an SIC scheme experiences in MC CDMA due to phase and timing tracking errors. It is shown that SIC can still retain a performance advantage over MF-MRC, although the advantage of SIC decreases with increasing tracking errors, especially when subcarrier correlation coefficients are small.