I've found a few cases where they've been quite handy:
Management of complex private state, such as an InterpolationTriangle used by an Interpolator class. The user of the Interpolator doesn't need to know that it's implemented using Delauney triangulation and certainly doesn't need to know about the triangles, so the data structure is a private nested class.
As others have mentioned, you can expose data used by the class with an interface without revealing the full implementation of a class. Nested classes can also access private state of the outer class, which allows you to write tightly coupled code without exposing that tight coupling publicly (or even internally to the rest of the assembly).
I've run into a few cases where a framework expects a class to derive from some base class (such as DependencyObject in WPF), but you want your class to inherit from a different base. It's possible to inter-operate with the framework by using a private nested class that descends from the framework base class. Because the nested class can access private state (you just pass it the parent's 'this' when you create it), you can basically use this to implement a poor man's multiple inheritance via composition.