The purpose of this study is to clarify the role of suppression in the growth of masking when a signal is well above the masker in frequency (upward spread of masking). Classical psychophysical models assume that masking is primarily due to the spread of masker excitation, and that the nonlinear upward spread of masking reflects a differential growth in excitation between the masker and the signal at the signal frequency. In contrast, recent physiological studies have indicated that upward spread of masking in the auditory nerve is due to the increasing effect of suppression with increasing masker level. This study compares thresholds for signals between 2.4 and 5.6 kHz in simultaneous and nonsimultaneous masking for conditions in which the masker is either at or well below the signal frequency. Maximum differences between simultaneous and nonsimultaneous masking were small (<6 dB) for the on-frequency conditions but larger for the off-frequency conditions (15-32 dB). The results suggest that suppression plays a major role in determining thresholds at high masker levels, when the masker is well below the signal in frequency. This is consistent with the conclusions of physiological studies. However, for signal levels higher than about 40 dB SPL, the growth of masking for signals above the masker frequency is nonlinear even in the nonsimultaneous-masking conditions, where suppression is not expected. This is consistent with an explanation based on the compressive response of the basilar membrane, and confirms that suppression is not necessary for nonlinear upward spread of masking.