Here's my take on it all:
Typical numbers are the neck pocket is 5/8 deep. A neck is 1" thick. The neck sticks up 3/8". You can rout the neck pocket down. You can remove material ( maybe , depending on truss rod anchor) off the bottom the neck. You could put some material under the bridge to elevate it. Some people would recommend putting a shim at the front of the neck pocket at an angle, making the neck tilt up... which I think is bogus.
My logic is this. Which has more value and is more accurate? If it is a 30 dollar neck from China and the neck is too thick, I'd take it off the bottom of the neck. If the error is in the body, I'd fix the body. That is assuming your saddles won't go up that 1/8" or so to give you the correct action you like.
To remove the material from the neck, I'd scribe a line around it with a combination square and awl, and probably sand it on a belt sander, ensuring I don't go past the scribed line.
If I'm removing material from the pocket, I'd get a router with a large base, and use a short pattern bit to ride inside the pocket. It can also be done with some chiseling and router planing ( both hand tools) too.