Inner class within Interface

A valid use, IMHO, is defining objects that are received or returned by the enclosing interface methods. Tipically data holding structures. In that way, if the object is only used for that interface, you have things in a more cohesive way.

By example:

interface UserChecker {
   Ticket validateUser(Credentials credentials);

   class Credentials {
      // user and password
   }

   class Ticket {
      // some obscure implementation
   }
}

But anyway... it's only a matter of taste.

Yes, you can create both a nested class or an inner class inside a Java interface (note that contrarily to popular belief there's no such thing as an "static inner class": this simply makes no sense, there's nothing "inner" and no "outter" class when a nested class is static, so it cannot be "static inner").

Anyway, the following compiles fine:

public interface A {
    class B {
    }
}

I've seen it used to put some kind of "contract checker" directly in the interface definition (well, in the class nested in the interface, that can have static methods, contrarily to the interface itself, which can't). Looking like this if I recall correctly.

public interface A {
    static class B {
        public static boolean verifyState( A a ) {
            return (true if object implementing class A looks to be in a valid state)
        }
    }
}

Note that I'm not commenting on the usefulness of such a thing, I'm simply answering your question: it can be done and this is one kind of use I've seen made of it.

Now I won't comment on the usefulness of such a construct and from I've seen: I've seen it, but it's not a very common construct.

200KLOC codebase here where this happens exactly zero time (but then we've got a lot of other things that we consider bad practices that happen exactly zero time too that other people would find perfectly normal so...).

Yes, we can have classes inside interfaces. One example of usage could be

public interface Input
{
    public static class KeyEvent {
         public static final int KEY_DOWN = 0;
         public static final int KEY_UP = 1;
         public int type;
         public int keyCode;
         public char keyChar;
    }
    public static class TouchEvent {
         public static final int TOUCH_DOWN = 0;
         public static final int TOUCH_UP = 1;
         public static final int TOUCH_DRAGGED = 2;
         public int type;
         public int x, y;
         public int pointer;
    }
    public boolean isKeyPressed(int keyCode);
    public boolean isTouchDown(int pointer);
    public int getTouchX(int pointer);
    public int getTouchY(int pointer);
    public float getAccelX();
    public float getAccelY();
    public float getAccelZ();
    public List<KeyEvent> getKeyEvents();
    public List<TouchEvent> getTouchEvents();
}

Here the code has two nested classes which are for encapsulating information about event objects which are later used in method definitions like getKeyEvents(). Having them inside the Input interface improves cohesion.