I know this question is really old but I would just like to add my take on partial classes.
One reason that I personally use partial classes is when I'm creating bindings for a program, especially state machines.
For example, OpenGL is a state machine, there are heaps of methods that can all be changed globally, however, in my experience binding something similar to OpenGL where there are so many methods, the class can easily exceed 10k LOC.
Partial classes will break this down for me and help me with finding methods quickly.