Science News carries a report on How Spaghetti Breaks.

The breaking of Spaghetti is a curious phenomenon. For a casual cook like me, the passing notice that the handful of spaghetti used to make dinner won't break in half perfectly when bent is usually attributed to uneven-ness of the force applied or unsteady hands. For a serious scientist like Feynman, however, the fairly verifiable observation is just too much. I quote from the article:

In the midst of making a spaghetti dinner for themselves one night about 20 years ago, Feynman and a friend--supercomputing innovator W. Daniel Hillis--launched into a brief investigation of this perplexing breaking-pasta performance. "We ended up, at the end of a couple of hours, with broken spaghetti all over the kitchen and no real good theory about why spaghetti breaks in three," Hillis recalls...

It seems that puzzle might be solved now. Experimental physicists and applied mathematicians in French and US produced two separate models for the fragmentation of brittle rods.

There was first the work of Gladden, Handzy, Belmonte, and Villermaux ("Dynamic buckling and fragmentation in brittle rods", Phys. Rev. Lett. 94, 35503 2005), which by experimental methods with a high-speed camera, captured the buckling phenomenon of a brittle rod compressed at both ends in situ. (See the science news article, or read the Phys Rev Letter for more details.) Though from the summary it seems that they only experimented with an impact object at speeds below the speed of sound propagation in the brittle rod. It would be logical to expect, however, that should an object impacts the rod on one end and speed exceeding speed of sound in the object, some sort of shockwave would form, and for one-dimensional brittle media, the rod will just crumble until it absorbed enough energy from the object to make the impact sub-sonic.

Then there was the work of Audoly and Neukirch ("Fragmentation of brittle rods: why spaghetti do not break in half", Phys. Rev. Lett. 95, 095505 2005). This study differs in that the spaghetti are bent rather than compressed on ends (the authors argue that most people don't break their spaghetti by pushing on it very hard on the ends in the kitchen). Solving a non-linear Kirchoff equation (or rather, numerically approximate the solution of such an equation), they get a fairly accurate model of how a wave traverses the length of the spaghetti subject to one Dirichlet and one Neumann boundary condition. Experiments verify their claim that spaghetti breaks when the curvature exceed a certain threshold. Check out their webpage, it has nice movies of the experiments and simulation.

Update (Nov. 20): The ever-so-lovely S points out that obviously, the above research applies to other long, thin objects, and explains why fencing foils tend to also break in 3+ pieces.