It's true that band-limited signals can't be time-limited (and vice-versa). However, band-limited and time-limited in that context mean "having finite support", i.e. the respective function is identically zero outside a given interval.
In practice, we almost always deal with functions that are time-limited and band-limited, because we never have infinite time and infinite bandwidth available. However, in that context we use the terms "band-limited" or "time-limited" in an approximative sense. I.e., the contribution of the function outside a given interval becomes negligibly small compared to the noise in that interval. We don't require it to be exactly zero.
Using your example of human speech: if you utter a sentence, no matter how long it is, it will be finished at a certain moment. However, the acoustic signal you've produced will be reflected and reverberated, and it will decay exponentially. Soon the reflections will be inaudible and will be hidden in noise, but in theory they go on forever.
When you say that human speech is band-limited to $4$ kHz, then that's actually not true. We often use that frequency as a cut-off frequency because we know that speech remains intelligible if we filter out frequencies above $4$ kHz. Yet, there is a significant contribution above that frequency, and the harmonics continue indefinitely. But again, their contribution becomes negligible and they disappear in noise.