Use number 3:- Better maintenance by compacting large classes
The other important use of partial classes is for better maintenance of the project. If you have large classes with many methods as shown in the following figure , it's a bit of a pain to maintain those classes.
By using partial classes you can split them into physical files as shown in the following figure, thus improving your project and making it easier to maintain.
Use number 4:- Multiple people working on the same class
The last and final real time I see of partial classes is when we want two developers to work in the same class simultaneously. I agree this can be a very rare use as there are better options like using a version control software like TFS or Sub version, but in case you want something quick and local this option is not bad at all.
You can also watch my 500 videos on various technologies like .NET, C#, Silverlight, Azure, VSTS, WCF, WPF, WWF, Share Point, design patterns, UML and much more.