Regardless of the RNP of a particular route or airway, these days ATC expects you to be ON CENTERLINE unless cleared otherwise (e.g., SLOP or other offset, or weather deviation). Effectively, the "acceptable" lateral deviation is ZERO, or the resolution of the ATC radar.
Best example I can think of is to try a 1 mile deviation just after takeoff on an RNP 1 SID. Think you'll win any argument based on the RNP1 designation?!?