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.
Il vaut mieux hasarder de sauver un coupable que de condamner un innocent.