There are two variants, one happens on hiding the popup the other on moving the mouse over the arrow. Below is a minimal example to reproduce both:

Variant A:
- run the example as-is
- open popup
- hide popup and see the NPE

top of stacktrace:

Exception in thread "JavaFX Application Thread" java.lang.NullPointerException
	at javafx.scene.control.skin.ComboBoxPopupControl.lambda$createPopup$913(ComboBoxPopupControl.java:464)
	at javafx.stage.PopupWindow.doAutoHide(PopupWindow.java:840)

Variant B:
- uncomment the line setting editable to true, compile and run
- move mouse over arrow and see NPE

Top of stacktrace:

Exception in thread "JavaFX Application Thread" java.lang.NullPointerException
	at javafx.scene.control.skin.ComboBoxBaseSkin.lambda$new$900(ComboBoxBaseSkin.java:108)
	at com.sun.javafx.event.CompositeEventHandler$NormalEventHandlerRecord.handleBubblingEvent(CompositeEventHandler.java:218)

The technical reason for both is the access to the behavior via getBehavior which is package-private (thus can't be overridden) and defaults to null.

A fix might be tricky: can't widen the method scope because Behavior is private api which must not leak into public api (I assume those are the rules?)

A working hack is twofold: 
- create our own popup (c&p core, replacing getBehavior by our own) and reflectively replace super's field value
- make sure we set editability only after the skin is installed 

The example: 

public class ComboCustomSkinNPE_Report extends Application {

    /**
     * Minimal custom skin and behavior.
     */
    private static class CustomComboSkin<T> extends ComboBoxPopupControl<T> {

        private Pane content;
        private Label display;
        private ComboBoxBaseBehavior<T> behavior;
 
        public CustomComboSkin(ComboBoxBase<T> control) {
            super(control);
            // most basic behavior
            behavior = new CustomComboBehavior<>(control);
            getPopupContent().setManaged(false);
            getChildren().add(getPopupContent());
            getChildren().add(getDisplayNode());
        }
        
        //---------- implement abstract methods
        @Override
        protected Node getPopupContent() {
            if (content == null) {
                content = new VBox(10);
                for (int i = 0; i <5; i++) {
                    content.getChildren().add(new Label("dummy-item " + i));
                }
                
            }
            return content;
        }

        @Override
        protected TextField getEditor() {
            return null;
        }

        @Override
        protected StringConverter<T> getConverter() {
            return null;
        }

        @Override
        public Node getDisplayNode() {
            if (display == null) {
                display = new Label("nothing real");
            }
            return display;
        }
        
    }
    
    private static class CustomComboBehavior<T> extends ComboBoxBaseBehavior<T> {

        public CustomComboBehavior(ComboBoxBase<T> comboBox) {
            super(comboBox);
        }
        
    }

    private Parent getContent() {
        ComboBox customFromStart = new ComboBox(FXCollections.observableArrayList("one", "two", "three")) {

            @Override
            protected Skin createDefaultSkin() {
                return new CustomComboSkin<>(this);
            }
            
        };
        // if editable _before_ installing skin, throws NPE on moving mouse over arrow
        // customFromStart.setEditable(true);
        Pane coreLane = new HBox(10, new Label("combo core with custom skin"), customFromStart);
        
        return new VBox(10, coreLane);
    }
    
    @Override
    public void start(Stage primaryStage) throws Exception {
        primaryStage.setScene(new Scene(getContent()));
        primaryStage.show();
    }

    public static void main(String[] args) {
        launch(args);
    }
}