Any custom rifle that won't shoot .5 in whatever cartridge isn't worthy of the name. The good ones easily hit .2-.3 and that is my expectation of a custom. There is no such thing as an inherently inaccurate round but some rounds like the 338 WM and 338 RUM do seem to be easier to tune for some reason.
If a rifle won't put the first cold bore shot where is puts the warm bore groups, it isn't the right gun for hunting and it wasn't built right. That said, I have never seen a rifle that shot consistent cold bore POI/POA that didn't group well. A gun that shoots 1" or more at 100 isn't going to cold bore a 5" circle at 800 yards every time.