Are there are good uses of Partial Classes outside the webforms/winforms generated code scenarios? Or is this feature basically to support that?
LINQ to SQL makes good use of partial classes to extend designer generated code. I think you will typically find this pattern of partial classes being used by designer-created code.
Anywhere you'd have used #region
sections before probably makes more sense as separate files in partial classes.
I personally use partial classes for large classes where static members go in one file and instance members go in the other one.