Suppose we have a list $R$ of axis-aligned rectangles in the plane. There is the well-known problem of determining the maximum subset of $R$ which consists of disjoint rectangles; this problem is known to be NP-hard. There are reasonably efficient algorithms, however, for finding approximations to the maximal independent subset; e.g., see this paper.
I have a twist on this problem. I don't need the maximal independent subset or even a single approximation to it. Rather, I need a list of subsets $R_1,R_2,\dots,R_k$, each of which consists of disjoint rectangles, such that $R_1\cup R_2\cup\dots\cup R_k = R$. It is not necessary that the sets $R_1,\dots,R_k$ themselves be disjoint, though in an ideal solution they would be. I want to find such a list as quickly as possible, with the additional optimization constraint that each individual list should be large (so as to rule out the trivial solution where each list consists of a single rectangle).
If you're curious about the motivation of this problem, it arises in cartography when one wishes to dynamically label a map: each set of labelings corresponds to a set $R_i$ above, and the interactive map cycles through the label sets $R_i$ over time so that all labels are guaranteed to be displayed.
Another formulation of your problem: you have a (not neccessarily planar) graph (its vertices are your rectangles and two vertices are connected by an edge if these rectangles overlap) and are looking for a coloring of its vertices (each color being one of your subsets $R_i$) using a low but not neccessarily minimal number of colors.
There exist various algorithms for this, with different tradeoffs between speed and quality of the result. There is also plenty of papers written on this subject. The Algorithm Design Manual by Skiena suggests http://webdocs.cs.ualberta.ca/~joe/Coloring/ and http://mat.gsia.cmu.edu/COLOR/color.html as useful resources, and also gives help in choosing an algorithm as well as an overview of existing implementations. Among these implementations are GraphCol and GOBLIN. It also has plenty of suggestions for further reading.