Are there are good uses of Partial Classes outside the webforms/winforms generated code scenarios? Or is this feature basically to support that?
As mentioned earlier, I too think this is a code smell.
If a class is so big that it needs to be split into more files, means that it is breaking the single responsibility principle and doing too many things. The large class could be broken down into smaller classes that cooperate together.
If you have to use partial classes or regions to organize code, consider if they should be in their own classes. It increases readability and you'd get more code reuse.