This situation pops up quite regularly when a relatively simple feature such as the diameter of a hole in a machined part doesn’t appear to be right when the part is at the assembly stage of manufacture. Like similar disputes, the finger pointing begins and compromises are made but the problem doesn’t go away. A fixed limit plug gage rejects the hole while direct measurements with various instruments says it’s okay so a coordinate measuring machine (CMM) is brought into the fray and its numbers disagree with those obtained by the other two methods. But since the CMM is a sophisticated device and a lot more expensive, there’s a tendency to believe what its readings imply compared to the lowly plug gage. The battle lines get shifted to a fight between how good each device is as opposed to how precisely each method provides values for the functional size of the hole, which is what the problem was in the first place. 

A look at how each of the methods does what it is supposed to do is revealing and points the way to resolving the issues. My comments assume each device is calibrated and functioning well.