Well, starts, yes, but very slowly in general. It's a physical state change rather than a chemical one, from one allotrope to another. In this case from a silvery metal to a grey powdery solid.
Low temperatures are an important catalyst in this change, and the preferred range is -40C or lower, although it can be seen as high as 13-14 C (the process is usually incredibly slow at this temperature however, even though tin blight is a self-catalysing change). The presence of germanium catalyses the change, too. Most of us should be a lot more worried about the solder in our cars and other outdoor electronics than about our tin soldiers.
Still, storing beloved metal miniatures in a cold, damp room or garage is a bad idea.