Aside from the other answers...
I've found them helpful as a stepping-stone in refactoring god-classes. If a class has multiple responsibilities (especially if it's a very large code-file) then I find it beneficial to add 1x partial class per-responsibility as a first-pass for organizing and then refactoring the code.
This helps greatly because it can help with making the code much more readable without actually effecting the executing behavior. It also can help identify when a responsibility is easy to refactor out or is tightly tangled with other aspects.
However--to be clear--this is still bad code, at the end of development you still want one responsibility per-class (NOT per partial class). It's just a stepping-stone :)