Yes, a real common example is, on an event-based application, an object that has to do two (or more) things at the same time. For instance accessing a network service while tracking and external process or accessing a database.
You may be able to delegate most of the work needed to perform the subtasks to other objects (which would be unidimensional state machines). But there would be a point where you have to handle the interaction between them using a multidimensional state machine.
In any case, note that a multidimensional state machine can be converted into a unidimensional one if you flatten the state matrix using the cartesian product, or in practice, if you are able to organize the states in a tree (usually not all the multidimensional states may happen).