As TangoOversway already pointed out, the humans in Planet of the Apes did have the capacity for speech (i.e. making noises with their mouths). This is illustrated by Nova shouting Taylor's name. However, it should be noted that a loss of the capacity for human language is quite plausible when you consider the phenomenon of feral children:
They often seem mentally impaired and have almost insurmountable trouble learning a human language. The impaired ability to learn language after having been isolated for so many years is often attributed to the existence of a critical period for language learning, and taken as evidence in favor of the critical period hypothesis.
So, if the apes isolated human children during the first few critical formative years of their life, it would be nearly impossible for them to subsequently acquire any significant degree of language. This would not be due to a lack of the ability to speak, since they would still have vocal cords, etc. It would be a result of underdevelopment of the cortical brain regions providing support for understanding and producing language. In other words, they would be able to make or mimic noises, but wouldn't really have much understanding at all of what those noises mean.
It's hard to say exactly why the humans in Planet of the Apes didn't speak much. Perhaps they had been so thoroughly conditioned by the fear of the repercussions for speaking (i.e. punishment by the apes) that they just never did. Or perhaps many or all of them were "feral" and incapable of language, with Nova being an exception in that she understood the concept of identifying individuals by spoken names. Either way, the loss of the capacity for human language is plausible given what we know from modern neuroscience.