The patent interference procedure pops up in some unlikely places. It is a key plot point in Aaron Sorkin's play, The Farnsworth Invention, which deals with who invented television. And in 1962, Richard H. Stern, Justice White's first law clerk, pointed out in a law review article the logical paradoxes that arise when applying patent interference law.
You can get a sense of the process by looking at the graphic to the left, which illustrates what happens when two patent applicants A and B provoke an interference with patentee C. The procedure involves scads of motion practice, testimony and multiple levels of review, all revolving around the abstract notion of determining "who thought of that first?" I always enjoyed my dealings with interference issues and I will miss this part of patent law.