No, even if you sum any number of signals, as long as each component signal is bandlimited then the resulting signal is also bandlimited and can be reconstructe

No, even if you sum any number of signals, as long as each component signal is bandlimited then the resulting signal is also bandlimited and can be reconstructed perfectly according to the Sampling Theorem. This may be counterintuitive, but it's provable. (I could prove it using just what I remember from a couple of semesters of undergrad signal processing courses.) Of course a low sampling rate can result in errors, but what the sampling theorem says is that we can calculate exactly how high a rate we need to always perfectly reconstruct signals in a given band. This doesn't change just because your signal is constructed from many component signals. Of course, if you think the math is wrong, then you have an easy way to disprove it: Construct a counterexample out of "many waves interacting" and show that the resulting signal isn't perfectly reconstructed as the Sampling Theorem claims. (Or perform a double-blind test that shows any human can detect the difference between a 192 kHz recording and the same recording downsampled to 48 kHz.)