The transformer-coupled tube reverb driver has a problem with clipping at low frequencies and at high frequencies. You can see why this is by drawing some load lines. At low frequencies when the AC load impedance is small, the AC load line is so vertical that it prematurely hits the zero current line resulting in clipping on that side of the signal. At high frequencies when the AC load impedance is high, the load line is so horizontal that the other side of the signal clips over by the 0V grid line. So it's like the low frequencies clip because the driver runs out of current and the high frequencies clip because the driver runs out of voltage.
The complaints you hear all the time are that low frequencies make the reverb sound "muddy" and high frequencies make it sound "shrill". Maybe it's the distortion from the driver that is causing the objectionable sounds at those frequencies rather than the mere presence of those frequencies. In any event, many driver designs attack the problem by cutting the high and the low frequencies. The classic Fender reverb driver rolls off highs all by itself, but CBS saw fit to roll off highs even more by placing a capacitor between the plates and cathodes.
Anyway, those harmonics can make themselves heard to the listener if given the chance and I, amongst others, don't want them to.
It's all subjective, but maybe a piccolo could benefit from a little reverb?