Well, if you are trying to do this on a tight budget, it could be horribly difficult, bordering on impossible. However, if you are absolutely willing to spend what is likely to be extra funds to get exactly what you seek, then it's absolutely possible to do so... for Smith & Wesson especially but also for Colt guns, start shopping guns where the owner has requested, paid for and received a factory letter.
The letter isn't an insignificant amount of money and it also took someone time to request and wait for it. The gun is typically going to cost more with a factory letter than a similar gun without one.
There are many many folks with these on the Smith & Wesson forum.
What's a much tougher one to get (but some persistent folks still claim to try to chase them, with what I imagine is a VERY low rate of success...) are the guys who chase after an actual "birthday gun." They seek to own a gun that was shipped on their actual day of birth.
That's a major crapshoot, you've got to get awfully lucky to find something like that.
I'd estimate that it may be some 365 times more difficult than what you seek!