Many years ago I was a part-time commercial artist for a while, and I hung around with the printers a little to get the job run right, and picked up a little bit of printing knowledge along the way.
I used to ask the printers "Can this be done?" And their standard response was "We can do anything if you can afford it." Then they would suggest the best looking, cheapest price method for the particular job.
Four color offset is a printing method that is extremely fast for huge runs. For example Web presses can run press speeds can reach up to 50,000 impressions per hour. An impression is equal to one full press sheet (38 inches x 22 and three-fourths inches), which is 12 pages. That's six hundred thousand pages an hour. That is how Newsweek or Time can print millions of magazines in a week. because the plates do not touch the paper they do not wear out after so many impressions. I have seen web presses running so fast that the friction of the paper running through the presses creates so much heat that the speed is limited by the kindling temperature of the paper. The humidity is controlled exactly in big offset printing houses to allow for perfect paper/ink/press speed happiness.
However, the time and cost associated with producing offset plates and the offset printing press setup makes smaller quantity printing jobs impractical. As a result, smaller printing jobs are now moving to digital offset machines.
Since four color offset printing is designed for speed in producing large runs that is what they should be used for.
You would lose speed, and increase costs if you had to remove and reload the paper for each color. And re-registration (lining up the paper exactly so the tiny dots match up perfectly) would be a nightmare task.
Printing each color sequentially in a very short run is better suited to say, silk screen, or lithograph.
More and more printing stuff is just going to high speed digital copy machines where the limited size of the small run is immaterial to the set-up costs. Artists can do a lot of the set up stuff on computers that we designers used to have to depend on the printers to do. It gives us more immediate creative control and freedom.
Short four-color offset printing video:
http://computer.howstuffworks.com/offset-printing.htm
Color silk screen printing:
http://en.wikipedia.org/wiki/Screen-printing
Lithography:
http://www.lib.udel.edu/ud/spec/exhibits/color/lithogr.htm
Color Lithography in art prints and posters:
http://www.art.com/asp/customerservice/limitededitionfaq-asp/_/posters.htm
So, OhWhyDontIknowMath, in theory you could print one color offset, then remove the paper after the first color, reload the paper, (being careful to get prefect alignment for perfect color registration) Then print the second color, remove, reload, register and run again for the third color, then print, remove, reload, register and run again for the last color.
Too labor intensive, expensive, and slow. Do the math!
You'll see that that would defeat the advantages of four color offset machines that are built for super high speed printing of huge runs at amazingly fast speeds.
Here is a quick, down and dirty explanation of what happens and why:
The colors of the image are photographically "filtered" into tiny dots (called "benday dots.)" See the explanation here:
http://en.wikipedia.org/wiki/Halftone
These benday dots line up in perfect "registration" (alignment) and the eye blends them together to give the illusion of a continuous color tone in the printed picture.
Different kinds of printing techniques have different advantages and disadvantages. So you have to pick the printing process that best meets the particular demands of the specific job.
Could a race car haul dirt?
Yes, but not as well as a big and slow, pickup truck.
Match the needs of the job with the capabilities of the method.