You can try it with your own uncooked spaghetti if you want; it almost always breaks into three when you snap it. I am asking for a good physical theory on why this is along with evidence to back it up. Or, a reference to a good study previously done on this would also be satisfactory. The math behind everything would be a great bonus if it exists.
My hypothesis is that a strand of dry spaghetti has two main pressure areas being put on it due to how brittle and how high the length to thickness ratio is, and that they snap at almost the same time due to the manufacturing process of the strand causing it to have a very consistent breaking point. Also that vibrations from a first break could influence the causing of another. I'm not sure how this could be tested, though.
Answer
The breaking of dry spaghetti was discussed in a 2005 Phys. Rev. Lett. by French physicists Audoly and Neukirch. Bottom line is that elastic (flexural) waves propagating along the spaghetti cause local increases in curvature leading to multiple breaking points: abstract to article. In essence, your assumption "that vibrations from a first break could influence the causing of another" is correct.
This work earned both authors the 2006 Ig Nobel prize for physics.
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