FULL PRODUCT VERSION :
java version "1.8.0_112"
Java(TM) SE Runtime Environment (build 1.8.0_112-b15)
Java HotSpot(TM) 64-Bit Server VM (build 25.112-b15, mixed mode)

ADDITIONAL OS VERSION INFORMATION :
Microsoft Windows [Versión 10.0.15063]

A DESCRIPTION OF THE PROBLEM :
When you select a date with the format: dd/mm/yyyy or mm/dd/yyyy in the DatePicker's PopUp, and then click on the fourth position of the editor: dd/m(here)m/yyyy or mm/d(here)d/yyyy, and finally press the "delete" key the caret is placed in an incorrect position.

REGRESSION. Last worked in version 8u151

ADDITIONAL REGRESSION INFORMATION:
java version "1.8.0_112"
Java(TM) SE Runtime Environment (build 1.8.0_112-b15)
Java HotSpot(TM) 64-Bit Server VM (build 25.112-b15, mixed mode)

STEPS TO FOLLOW TO REPRODUCE THE PROBLEM :
Run DatePickerTest.
Open the DatePicker popUp.
Select a date "dd/mm/yyyy" or "mm/dd/yyyy".
Click on the fourth position of the editor: dd/m(here)m/yyyy or mm/d(here)d/yyyy
Press the delete key.

EXPECTED VERSUS ACTUAL BEHAVIOR :
EXPECTED -
The caret located in the fourth position: dd/m(here)/yyyy or mm/d(here)/yyyy
ACTUAL -
The caret located in the third position. dd/(here)m/yyyy or mm/(here)d/yyyy

REPRODUCIBILITY :
This bug can be reproduced often.

---------- BEGIN SOURCE ----------
import javafx.application.Application;
import javafx.scene.Scene;
import javafx.scene.control.DatePicker;
import javafx.scene.layout.GridPane;
import javafx.stage.Stage;

public class DatePickerTest extends Application {

    @Override
    public void start(final Stage primaryStage) {

        final GridPane gridPane = new GridPane();

        final DatePicker datePicker = new DatePicker();

        gridPane.add(datePicker, 0, 0);

        final Scene scene = new Scene(gridPane, 700, 700);
        primaryStage.setScene(scene);
        primaryStage.show();
    }

    public static void main(final String[] args) {
        launch(args);
    }
}
---------- END SOURCE ----------

CUSTOMER SUBMITTED WORKAROUND :
datePicker.getEditor().skinProperty().addListener(e -> {
    final TextFieldSkin skin = (TextFieldSkin) datePicker.getEditor().getSkin();
    skin.forwardBiasProperty().addListener((ov, o, n) -> {
        skin.setForwardBias(true);
    });
});