I have a file with nested classes, but it gets long enough to be unreadable. Is there a way to break nested classes into separate files?
Use the right tools.
I'm afraid not.Alternatively, you might consider converting some of them into simple Java classes: although nested classes increase encapsulation, you can certainly create a good OO design without them.
My two cents.
Write unit tests to cover the code you are about to refactor.
Then use refactoring tools, for example. Eclipse / JDT to retrieve each class. Run tests for each refactoring job.
For added pleasure, you can do this without unit tests.