Finally, a good question!
Many times we don't care about distortion as long as the meaning of the message isn't lost. Have you ever wondered why telephone audio quality sucks compared to CD quality? Telephone audio quality has been severely bandlimited in a very careful way to trade off quality for cheaper and easier transmission. However, the bandlimiting has been carefully designed for the human voice so legibility is not lost.
But yes, you're right in that we can't transmit a finite-time signal (which has infinite bandwidth) over a finite bandwidth (because this requires the signal to go on for infinite time). It's contradictory. Does that mean that only one of these kinds of signals can exist? That is actually the case.
One thing you slowly learn is that all these ideal mathematical principles you learn in class don't work out in real life when you put everything together. The math principles are ideal and rigid and like to treat the extreme cases of a single real-world phenomena as separate things (for ease of analysis). But in the real world, it's usually just one phenomena at work.
That means that one of our two signals is an idealization. One actually exists, and the other is the ideal approximation of an extreme case so we can analyze it more easily. So which can't exist? It shouldn't be too hard to see why an infinite time signal can't exist since no signal lasts forever. That would imply that no signal is truly bandwidth limited (which isn't so obvious). From that reasoning that means all signals are finite-time (which is pretty easy to understand) and infinite bandwidth (which can be rather puzzling).
So how does this all work then? Remember how filters can't be made that have perfect brick-wall roll offs. This is intimately tied into the fact that only infinite bandwidth signals exist while finite bandwidth signals do not. It means that all signal actually take up infinite bandwidth. It's just how fast the frequency components of the signal asymptotically decay towards zero. Using a filter will accelerate this. If a signal's frequency components decay fast enough, it won't interfere too much with other signals that mainly consist of other frequency components (which will be much stronger at those frequencies and overwhelm the residual frequency component of the first signal).
Saying that all real signals are finite-time (which any layman would agree with) is the same as saying all signals are infinite-bandwidth. It's also the same as saying that ideal brick-wall filters cannot exist. Are you able to wrap your head around that?