To reproduce run following code.
Open the http://download.oracle.com/javafx in browser.
Right click on "JavaFX 2.0 Beta documentation" image -> copy.
Press button "paste from clipboard" in sample application. You should see "JavaFX 2.0 Beta documentation" image.
Right click on "Oracle" image in browser-> copy.
Press button "paste from clipboard" in sample application. You should see "JavaFX 2.0 Beta documentation" image, but expected "Oracle" image.


import java.io.File;
import javafx.application.Application;
import javafx.event.ActionEvent;
import javafx.event.EventHandler;
import javafx.scene.Scene;
import javafx.scene.control.Button;
import javafx.scene.image.ImageView;
import javafx.scene.input.Clipboard;
import javafx.scene.input.DataFormat;
import javafx.scene.layout.VBox;
import javafx.scene.text.Text;
import javafx.stage.Stage;

/**
 *
 * @author Andrey Nazarov
 */
public class NativeClipboard2 extends Application {

    VBox contentSrc;
    VBox output;

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

    @Override
    public void start(Stage stage) throws Exception {
        stage.setScene(new Scene(contentSrc = new VBox(20), 500, 500));

        final Button source;

        contentSrc.getChildren().addAll(source = new Button("Paste from clipboard"));
        contentSrc.getChildren().addAll(output = new VBox());

        source.setOnAction(new EventHandler<ActionEvent>() {

            @Override
            public void handle(ActionEvent t) {
                
                Clipboard cb = Clipboard.getSystemClipboard();
                getClipboardData(cb);

            }
        });
        stage.show();

    }

    private void getClipboardData(Clipboard cb) {
        output.getChildren().clear();
        output.getChildren().add(new Text("Data formats:"));
        for (DataFormat type : cb.getContentTypes()) {
            output.getChildren().add(new Text(type.toString()));
        }
        if (cb.hasString()) {
            String str = cb.getString();
            System.err.println("###STRING: " + str);
// output.getChildren().add(new Label("PLAIN_TEXT:"+cb.getString()));
        }
        if (cb.hasUrl()) {
            System.err.println("###URL: " + cb.getUrl());
// output.getChildren().add(new Text("URL:"+cb.getUrl()));
        }
        if (cb.hasRtf()) {
            System.err.println("###RTF: " + cb.getRtf());
// output.getChildren().add(new Text("RTF:"+cb.getRtf()));
        }
        if (cb.hasHtml()) {
            System.err.println("###HTML: " + cb.getHtml());
// output.getChildren().add(new Text("HTML:"+cb.getHtml()));
        }
        if (cb.hasImage()) {
// System.err.println("###Image: " + cb.getImage());
            output.getChildren().add(new Text("Image:"));
            output.getChildren().add(new ImageView(cb.getImage()));
        }
        if(cb.hasFiles()){
            for(File f: cb.getFiles()){
                System.err.println("##File: "+f);
            }
        }
    }
}