It seems that with at least certain version of the Asterisk PBX, and the firmware I am running on my Aastra 6757i phones, the Aastra phone will try to negotiate a 16kbps version of G.711 (presumably as specified in G.711.1), and when Asterisks tries to send audio back to the phone, it isn't converted to an audible form.
I haven't investigated upon which end this problem exists (i.e. whether Asterisks is sending bad or no audio, or if the phone is interpreting it incorrectly). What I can see is that setting the codecs to basic (or manually specifying functional codecs) results in correct audio.
The light that burns twice as bright, burns half as long.