For sake of argument, couldn't FOIA requests be done to certain registry entries to see if some machine guns made after 86 are actually in the registry?
No, that the NFA registry is considered private tax payer records and are not subject to FOIA requests.
I'm talking about specific entries here. You might be able to do a FOIA for the "records of the records" that don't show any specific info on any specific person. Maybe something like the total number of records in the registry or something like that could e obtained from a FOIA. (Not that the ATF knows how many records are in the regisgtry).