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 lots of methods as shown in the below
figure , it's a bit pain to maintain those classes.

By using
partial classes you can split them in to physical files as shown in the below
figure thus making your project better and easy 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 simultaneously two developers to work in the same class. 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.
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