NAD 314 power cut out at higher volume of sound

One channel or both?

Based purely on what you say I would look for any dries and also intermittent failure of a transistor, typically drivers or hot running VAS stages... but it could be anything. You need to do some basic voltage checks when the fault appears. Check the supplies, all of them, and check the DC offset before the speaker relay under fault conditions,
 
Well I have NAD C320 which has random fluctuations in audio volume. However, it seems to be ok when running with the cover removed. I suspect that the +-18v regulators are beginning to fail. They run really hot because the input voltage is 37v, twice the load. Also the top cover is only partially vented. So there is too much voltage and not enough ventilation.

And the 2SD669A pass transistor is obsolete.
 
Check the power supply caps as well, and the regulation.
It may be simply thermal shutdown, if it happens after say 15 minutes at full volume.
That is a sign of ageing parts.
Like Mooly said above, check it when the fault appears, and if it is a random fault, which comes and goes, a headache to pin down...