I heard about this problem, now that Alex mentions it slightly differently (the original description of the problem was very confusing, KennyJ). It is called the Euler graph theorem since Euler first solved problems of this type. In memory of this problem, any path on a graph that uses all the edges only once is called an Eulerian tour, I believe.

Anyway, this is called the Konnigsberg bridge problem. Supposedly the story goes that there were two coasts and the island of Konnigsberg in between. Or maybe it was two islands...
If we call the top coast A, the bottom coast B, and the islands C and D, then there were two bridges from A to C, two bridges from B to C, one bridge from A to D, and one bridge from B to D. People were happy, until the architects built another bridge between C and D, since this disrupted people's walking patterns.

So the question is, is it possible to take a path which walks over every bridge exactly once? You can relate this to your rectangle problem by considering each box as an island and the segments as "bridges" separating the islands.