I think we have to be careful in how this is explained.
RNP is not a measure of how accurately the approach is flown. It is a measure of the probability that the FMC will be think it is somewhere that it isn't. As we get closer to the ground, we increasingly prefer that probability to be lower.
Consequently, the performance referred to in the terms "required navigation performance" or "actual navigation performance" does not refer to either the pilot's skill or the autopilot skill. It refers to the "ability to perform" of the complete navigational system. For example, the satellite data accuracy, transmission accuracy, and the receiving component's ability to interpret the data satisfactorily are all part of the system performance. Any breakdowns in the system function will create an alert that the ANP, actual navigational performance, is not acceptable for whatever RNP you are trying to use. Hence, if a satellite is missing, your flight plan may contain a warning of an ANP limitation around your ETA.
In other words, if a given satellite is off line, what is the probability that your FMC will think it is on centerline when it is more than 0.3 miles off centerline? If that probability, the ANP, is within the RNP, then the chances that the FMC is confused are within acceptable limits.
But you could be a half mile off track during the approach and you will not get an ANP alert. The rest of the on-board system may be screaming at you, as well as the other pilot, but the ANP will be just fine...the probability that the FMC is lost has nothing to do with your piloting ability.
The concept of RNP/ANP is totally dependent on average risk. The difficulty is that 99.9% of pilot training is predicated on specific risk. The average risk concept is pretty foreign to us, so the explanation of RNP can get convoluted, particularly after it has been re-written into the manuals six times over by flight ops/flight standards people of varying levels of understanding...