Partial classes would be useful, when:
1. Multiple programmers are working on same implementation class, each of the can code in separate file, separate set of methods (of same class) independently.

2. Large files can be splitted into multiple files. In C++ this can be achieved by "declaring" class in (small) header file, and one or more CPP files.

3. As Thread1 said, you can divide functionality, viz. UI, business logic, database read etc, into multiple .cs files. You can also place them into different directories.