Let me give you a visual representation of how CPI interacts with monitor resolution when it comes to 3D FPS such as Q3.
Let's draw in your mind the player model, see it from third person view for now.
Draw a big huge sphere around it, centered on the POV-camera (which is where the camera is fitted "into" the player model, so it's where you see the action from).
This sphere doesn't have a curved surface outside, but it's a field of points, the number of which depend on how "low" your in-game sensitivity is.
Each point represents seconds of an angle (or even lower) on the 3D space, so you will have a dot matrix in sphere shape surrounding the player model.
All those points are where the game "allows you to aim", or where the projectiles will be directed when you shoot.
Now, get a squared matrix where all the points are in an equal distance, with the number of dots per columns and rows equal to CPI.
Put that matrix on a plain surface, and "push" it towards the sphere until the extremes intersect.
So where your mouse aims is that squared matrix.
Then the game interprets that info and places the pointer on the closest point of the sphere.
As you can guess, depending on your in-game settings there are going to be more or less dots on the sphere, as this won't change in size, but it will change in dot density.
When it comes to the CPI matrix, it will get bigger, to the point that this precision will get lost on the sphere at some point.
Enter your computer monitor with it's resolution :
Project the sphere and the CPI matrix on your monitor's pixel matrix.
When you have "perfect pixel precision", it means that the CPI matrix, the projected sphere matrix and your monitor resolution are "overlapping", as in 1 pixel corresponds to only one CPI count dot, and that corresponds to one possible angle in-engine. (That's a lot harder to achieve than you might think)
If you have higher CPI than that and your game permits it, you can aim on a per sub-pixel basis, which will allow you to position your pointer in-game on a higher resolution than your projected sphere, but you will have no visual feedback. As in, you will move a tiny bit your mouse and "nothing will happen".
This can be "dangerous" for very long distance battles, as you can have your mouse pointing at one specific dot of the sphere and your visual makes you think it's a bit on the side, so you will get the feeling you are not "hitting" the opponent.
If you have lower CPI or your sphere provides far less dots (by setting an in-game sensitivity that's very high), you can get into dangerous waters again, as pixel skipping might happen, or stair movement will be apparent.
So, what to strive for?
Just get a compromise on what feels good for you, and where you have a balanced resolution, CPI and in-game settings, where you don't get pixel skipping and you don't need too much movement to move one pixel on-screen, and you can move 1x1 pixels without problem, all the time (consistency is key).
Before someone comes and tries to axe all this because it's wrong :
Yes, I know some parts of this are "wrong" and I'm not explaining in any way "in-depth", it's a way to graphically explain, more a less, what's going on.