One way people are beginning to probe the mechanisms of surface dynamics is to compare the histories of particular surface configurations obtained using microscopy with predictions from simulations. An advantage of this technique over the usual method of studying the time dependence of quantities averaged over the surface is that it allows the direct determination of how individual surface defects (such as steps) move in response to their local environment. A fundamental question about this approach is how well, and for how long, can one expect to account for the evolution of the complicated surface configurations which are typically observed in experiments. At small enough length scales, unpredictable thermal fluctuations begin to play a role. In this talk I discuss the basic theory and implications of the breakdown of deterministic equations of motion of surface morphology caused by these fluctuations. I will illustrate the issues involved. for several systems: First, given a configuration of surface steps, I will discuss how well one can predict where, and in what sequence, islands nucleate during epitaxial growth. I will compare theory with experiments of vacancy island nucleation during O etching of Si(001) which show that nucleation gets less predictable as the temperature (and critical nucleus size) gets smaller. Second, I will discuss how long one can in principle expect to account for the structure of adsorbed layers during domain coarsening and for island configurations during ripening. I will give theoretical and experimental examples of how sensitive histories of such complicated structures are to chance events and the knowledge of initial conditions.