Now that I think of it I wonder if my scope would have shown a lot of ripple? Was the voltage regulator shutting things down?
I don't think it was a ripple that was the problem as the 2.7 V is derived from a regulated 5 V.
I think it was more like a stability issue caused by too high ESR on a worn out cap, as suggested earlier in this thread.
The voltage regulator wasn't shutting down, but there may have been instability causing IC402 to malfunction.
Edit: I just noticed that the main power supply unit is not a linear job but an SMPS.
The presence of an inductor in the 2.7 V may be there to help block any remaining switching noise (linear regs, i.e. IC410, do not do a good job at filtering out higher frequencies).
Or maybe the noise from IC402 is kept from going back into the 5 V supply by that little inductor.