There are 2 different issues.

One is the possession of the gun with the altered,obliterated ser#,,and that is clearly an offense of Fed Regs (see the letter in kwill1911 post). Also same language in all 50 states laws.

The other is the application to the BATFE to allow a ser# be restamped that has previously been removed/altered/obliterated. Or sometimes to restamp a ser# in a different location on a firearm, such as in custom work. The first instance happens many times from refinishing and alteration work. Other removals/obliterations occur through simple ignorance but with no criminal intent.

If the BATFE determines that the gun is not 'contraband', then they may at their decresion allow it (remarking). They usually do in the case of the drilled through original number or buffed orig#, etc.
They're not real warm to the idea of moving a ser# to another location on the gun for the convenience of custom work in my past experiences.

They may also demand that a new, unique number be placed on the frame in place of whatever original serial number you may believe was there.
In those cases the new ser# will have an 'ATF' prefix and is most likely to be applied by a LE agency before returning the firearm to the rightful owner. This usually happens in cases of a recovered stolen firearm and then not very often. I've run accross only 4 in the last 40 years of working on them,,all handguns.

They're not hard to work with,,but they do have their regs to follow also.

You can do yourself a favor if you know the guns history too before starting down this road.
Restamping a missing number is one thing, but not knowing the reason it was removed in the first place is another. Some are obvious wear, gunsmithing work, etc, others bring suspicions of theft.