I just got a $200 check from the United States Treasury, and toward the bottom left it says "NFA Application refund".  I have all of my stamps back with approved Form 4s that I sent in thru April 2016 (pre-41P).  I did have a few Form 4s get kicked back to my dealer for not sending in two copies, two originals, etc, but all of those were corrected within a few days, sent back in, and eventually approved in 2016.  This seems like an error that the ATF initiated.  Do I deposit this or start calling the NFA Division?

just send it back to them... with another form.

A call might be in order.

FedGov does not forgive or forget erroneous refunds, and most likely will someday come back at you for the refund, plus interest, if you don't get it nipped in the bud.

I'd cash it

Cash it today, make them wait for 9 months to return it.

I like free government money as much as the next guy, but most likely one of your stamps got canceled by accident and is no longer valid.   So I would call them up and figure out what happened.

I'm going to cash it, but I'm going to go back thru my bank statements and look at all of my $200 checks going back 2 years since I started my pre-91F buying spree.  I now remember one application that was messed up by my dealer and not sent in, and they "lost" the check.  So, we completed a new Form 4 and enclosed another check.  I never cancelled that other check, since it was made out the the "Bureau of ATF".  I'm guessing it somehow got sent in, but with what, I'll never know.  And I'm guessing it got deposited.  I am not calling them.  I get way too frustrated trying to work out a government error or something that falls out of the normal process.