For clarity, there is "bore diameter" and there is "groove diameter". Bore diameter is the measurement within the rifling. Groove diameter is the measurement within the depths of the rifling grooves.
Manufacturing tolerances will have some effect. Tool wear within each production cycle may cause measurable differences in barrels produced by the same manufacturer using the same equipment and tooling.
SAAMI specifications prevail in the US for sporting arms produced by most companies. Those spec's call for "nominal" dimensions, with allowances for manufacturing tolerances.
Example: The .357 Magnum usually has a nominal bore diameter of .348" to .350" and a groove diameter of .356" to .358". But commonly used mass-produced revolvers by major manufacturers will frequently have bores from .344" to .352" and grooves from .354" to .360", and greater variations are known to exist.
Loading and shooting jacketed bullets generally allows for the use of bullets up to .001" over groove diameter without problems.
Loading and shooting cast or swaged lead bullets generally allows for the use of bullets up to .002" or even .003" above groove diameter without problems.
Best results are usually reported with bullets matching groove diameter, or slightly (.001" more or less) above groove diameter.
Other factors come into play. Chamber dimensions may not allow cartridges to be seated easily when loaded with bullets above certain dimensions. Cartridge cases may have unusually thick case walls that, when combined with slightly oversized bullets, may prevent proper functioning in a particular firearm. Barrel forcing cone dimensions (and angle of the forcing cone cut) can cause pressure problems when an oversized bullet is forced through under pressure of discharge.
The only way to be certain of a predictable result is to slug your new barrel and measure your chambers and forcing cone. This can result in some unhappy news (such as oversized chamber, undersized forcing cone, or other factors not compatible with the actual dimensions of the new barrel (bore and groove diameters). Such events are probably more likely to be encountered with a custom gun than most factory models.
Then there are variations in bullet dimensions (also subject to manufacturing tolerances). While the nominal dimension of a bullet might be .400" diameter, you are likely to experience variations from about .398" to .402", or perhaps wider ranges.
Cast lead bullets will vary depending on the mold, as well as upon the composition of the lead alloy used for casting. I have noted performance from the same mold (.358" nominal diameter) ranging from .356" to .360" just by going from wheel weights to linotype metal. Sizing cast bullets in the same .358" sizing die will yield differences from .357" to .360" depending on the casting metal alloy used (and sometimes the bullets will display some "spring back", measuring .358" on the day of sizing, but popping up to .360" a few weeks later).
Assuming that the groove diameter of your new barrel is actually .3948" as expected (highly unlikely, but possible) I wouldn't worry overly much about using lead bullets (cast or swaged) having a diameter of .400" or even .401" (always assuming a careful and conscientious application of loading standards working up the load from minimums). Even that won't guarantee good results, especially if the chamber dimensions won't allow your cartridges to fit properly, or your forcing cone won't accept the bullets easily.
Way too many factors involved to overly simplify such a discussion.
Have I confused matters enough yet?
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