To extend on the answer posted by dmckee, there are uniqueness theorems for Laplace's equation, Poisson's equation, the wave equation, and Schroedinger's equation (for the Hamiltonians of most simple physical systems), and of course more, though these are probably the ones you are referring to.
Another thing to keep it mind is that this does not imply solutions are necessarily not separable, as linear differential equations allow us to take sums of solutions and remain in the solution space. Thus functions like $x+y+z$ might still be solutions even though they are not separable, so long as they can be built out of separable solutions. I would venture to guess that the separable solutions constitute a basis for the solution space in just about all common cases.
Also have a look at this and this too.