There's a lot of discussion out there on this topic, and lots of people saying that 1) it's bad design to use partial classes, 2) that it's used for autogenerated code, and 3) that it shouldn't take the place of inheritance.
I have a situation, though, in which partial classes look like they'll come in very handy: I'm building a series of applications which will eventually be integrated into a suite. They'll all have a main form which will provide some functionality, and several shared components (e.g., a form to display reports). While I could define a base class and inherit from it, that would mean a lot of rework when the time comes to combine all of the applications into the "enterprise" version of the product.
Thus, partial classes are quite useful, because I can quite simply include the various partial classes into the combined version, while still allowing me to build the individual, stand-alone versions of the product. If I were to try to accomplish this using inheritance, I'd end up with each component calling its own version of the common components (e.g., InvoiceReportViewer, PurchasingReportsViewer, etc.) rather than simply calling ReportsViewer and knowing that Visual Studio will have integrated all of the bits for me.