You are correct that the bass does indeed require more power. So, if the amp were under-powering the speakers then that could explain the problem. BUT, that would not be a problem inherent with Class D topology but rather a problem of too little power (ie trying to drive the amp beyond its limits). Too little power can be a problem with ANY class of amp.
I'm sorry, I should be more clear and precise with high frequencies and Class D amps. Problems with the high frequency response WAS a problem at the birth of the class D topology. As far as I can tell, MODERN Class D amps have over-come those problems that the first few generations of class D amps suffered from.
So, HISTORICALLY there were problems with class D reproducing the high frequencies. Since Class D are "switching amplifiers", the "switching" needs to be orders of magnitude quicker than the highest frequency (20kHz), and it took a few generations of designs to get that figured out.
As far as Class D amps having problems reproducing bass (low frequencies), that has never been an issue associated with class D. Furthermore, class D amps are very popular in the car audio community and in the SPL competitions (driving subs).
On a side note, I'm danged interested in those new NAD Master Series that AH just announced. They are using Hypex nCore amps!