Multimode fiber-based saturable absorbers enable mode-locking in lasers, generating ultrafast pulses and providing an exceptional platform for investigating nonlinear phenomena. Previous analyses in the continuous-wave (CW) limit showed that saturable absorption can be obtained due to nonlinear interactions between transverse modes. We find experimentally that saturable absorption can be achieved, thanks to the interplay of single-mode fiber nonlinearity and the wavelength-dependent linear transmission of the multimode fiber, even with negligible intermodal nonlinearities. We further show that even when intermodal nonlinearities are significant, the CW analysis may not be sufficient for long multimode fibers. Understanding the underlying mechanisms of multimode fiber-based saturable absorbers opens new possibilities for developing programmable devices for ultrafast control.