I saw that episode too, and wondered if the person was a firearms appraiser or just an antiques person. There is more to deciding value, but maybe we did not see the entire thing (obviously the appraisal had already been done before the camera started rolling).
That being said, yes, you can obtain factory letters on certain firearms. I got one for my husband's AH Fox shotgun last year. I don't know if all the manufacturers do this, but essentially you pay someone to do the research in company records. Info included often covers name of the original purchaser, cost, placed shipped to, any special features (IE special inlays/engraving), factory specs--chamber depths, choke types etc.
These letters are really very fascinating. The ones I have seen range in price from 60 to 150 dollars.
Check this link for more information. http://www.bbhc.org/firearms/records.cfm
Also, here is a gun for sale that lists the Cody letter in the description:
For those of you with older model guns (particularly shotguns) its a nice way to get some history on it. I have our letter framed on the wall.
Whether or not this letter would increase the value that much I don't know.....hmm.....I am not an expert in appraising, but it does make me wonder...
It did not seem to help the gun for sale on gunbroker.....