Consider a class which implements a lot of interfaces, would it make sense to implement each interface in a separate file using partial class definitions?
Would this be an abuse of the language feature or is it an idiom I'm unaware of?
If your class has to implement many interfaces, that's a reasonable way of managing the source, yes. You can edit the project file to make several of them depend on one "main" class file, which makes the Solution Explorer easier to work with.
You should ask yourself whether you shouldn't have several smaller classes each implementing a single interface though. Sometimes that will be a better approach, sometimes not - but it's always worth asking the question.