EDIT: I rewrote this first part for clarity.
If you are a beginner and you are watching a skilled artist draw, you will probably notice that they are skilled at creating depth, yet they are not using complex perspective construction to do this. They are just freehanding and drawing as if they are almost 3d modelling. Sculpting form with lines or tone.
I think that this is a key point that isn't usually stressed, but from talking to a lot of artists, there is a certain sort of illusion that the artist is immersed in while drawing. The effect is somewhat similar to putting on 3d glasses, where you pen feels like it doesn't just have an x and a y axis but also a Z.
When you look at a terrible artist, one who obviously has put in a lot of time and their drawings just look wrong, chances are that they have problems drawing form and it can likley be taken back to the lack of this sensation, and their drawings as a result look really bizarre.
Now it's one thing to explain the experience, and why it is important but i think it is possible to evoke it in an artist through exercises. Below is my attempt at writing one and it seems to be effective on the artists i have tried it on.
You are to draw a perspective scene with a cube, and this is to set up the illusion. Then you are tasked with drawing primitives floating in the illusion. Some should be really close to the camera, others should be far away. Most should be rotated at arbitrary angles. These primitives are to be eyeballed, without planning, as we are testing to see if you can draw within the illusion.
Pay attention to the original cube as that is defining the perspective for your primitives.
Start out with cubes. Once you feel that you can draw them in the space perfectly move up to cylinders, cones, triangular prisms, pyramids.
The next step up in difficulty is to drop the cube construction which you used to define the perspective, and instead use an axis drawing to define it. So you are dropping out information to make sure you can picture the perspective with the minimum amount of information.
Once you can comfortably do this with primitive objects, start moving up the complexity of the objects, depending on the level of drawing skill you possess.
This will be very difficult if you are a beginner artist, but i think it's something that everyone can tackle if you put your mind to it.
So this exercise is also designed to do a few things other than get you into that illusion. It is designed to ramp up difficulty in a very deliberate way so that you can identify where you begin to have problems with form, so that you can stay at that level until you beat it.
Think of it like weight training. If you keep trying to pick up weights you can't comfortably get off the ground you won't get anywhere, and the same with weights that are too light, you need to find weights just above where is comfortable and push the edges so that you can still use clean technique.
Let me know if this is helpful and please post up your results if you try this. If it's helpful i might make this a series of posts on fundamental skills.