Cochlear implant (CI) listeners typically perform poorly on tasks involving the pitch of complex tones. This limitation in performance is thought to be mainly due to the restricted number of active channels and the broad current spread that leads to channel interactions and subsequent loss of precise spectral information, with temporal information limited primarily to temporal-envelope cues. Little is known about the degree of spectral resolution required to perceive combinations of multiple pitches, or a single pitch in the presence of other interfering tones in the same spectral region. This study used noise-excited envelope vocoders that simulate the limited resolution of CIs to explore the perception of multiple pitches presented simultaneously. The results show that the resolution required for perceiving multiple complex pitches is comparable to that found in a previous study using single complex tones. Although relatively high performance can be achieved with 48 channels, performance remained near chance when even limited spectral spread (with filter slopes as steep as 144 dB/octave) was introduced to the simulations. Overall, these tight constraints suggest that current CI technology will not be able to convey the pitches of combinations of spectrally overlapping complex tones.