According to the non-canon Ships of the Star Fleet: Volume I (the bit that had all those fold-out schematics of the Belknap-class strike cruiser), the Constellation was refitted from an older class (Horizon-class, if I recall the text correctly) of cruiser to Constitution specs before it was destroyed. There was a lot of that inexplicable stuff - like the 1600 registries when the (generally-accepted) registries for the Constitution class start with the Constitution herself, at 1700.
Edit to add something: After checking shipschematics.net, I found a design for the Horizon-class, which appears to be a refitted Daedalus...the registry is NCC-1000, which would fit the Constellation...the design gives me pause, but given enough tinkering, you could get a Connie out of it. Here is the schematic: http://shipschematics.net/startrek/images/federation/cruiser_horizon.jpg
As great as SOTSF is, it isn't even internally consistant, Volume II #1 lists NCC-1017 as an Akyazi class PA, USS
Abreus. But I generally follow SOTSF's theory regarding the
Constellation being refit from a Horizon class, it fits a lot better than most other explanations.
As far as the 16xx registries go, such as the
Farragut, however, those don't appear on screen until TOS(R), and they were made up by Greg Jein, as far as I'm concerned the
Farragut was a Connie class with a registry in the proper 17xx range, as per the SFTM, the SOTSF, Starship Farragut, and a whole ton of other sources.