In typical PCM recordings, like CDs, mid-range frequencies (e.g. 1kHz to 4kHz) are recorded with lower amplitudes because our ears are more sensitive to them.
Sampling theory is correct and 16-bits can reproduce any waveform with ~100dB of range, however, in a complex waveform consisting of low, mid and high frequencies, the mid- and hi-range frequencies quite simply get shortchanged.
Imagine a recording of a bass sinusoid and a mid-range sinusoid of equal volume. It might use e.g. 10 bits to store the bass and only 6 to store the high frequencies. (2^10sin(200wt)+2^6sin(4000wt)). That means the resolution of the high frequencies is less than the lower frequencies. When the volume of those frequencies changes dynamically, the high frequencies' amplitudes are more quantized. That is quite simply why 16-bits are not enough.
This is similar to the problem with storing waveforms unprecompensated on vinyl. The precompensation makes up for the non-uniformity of the medium. It could be done with 16-bit digital as well. Or alternatively, larger sample sizes like 24 can be used.
I haven't A/B tested this. The A/B test in the article compares CD with SACD. SACD isn't PCM, so its artifacts are going to be totally different from 24-bit PCM.