DATE_FIELD is not properly found in a short, medium, long or full date.
The following program is evidence.
It builds an instance of FieldPosition called dayPos based on DATE_FIELD,
then calls format(date, strbuf, dayPos) to print the date, but beginIndex
and endIndex of the field position are both mistakenly 0.
This example works for both month and year, but not for date (which is the
day of month, really).

import java.text.*;
import java.io.*;
import java.util.*;

class Main {
     // Create field position objects for dates.
     static FieldPosition dayPos = new FieldPosition(DateFormat.DATE_FIELD);
     static FieldPosition monthPos = new FieldPosition(DateFormat.MONTH_FIELD);
     static FieldPosition yearPos = new FieldPosition(DateFormat.YEAR_FIELD);

     // Create date formats.
     static DateFormat shortDateForm =
           DateFormat.getDateInstance(DateFormat.SHORT, Locale.US);
     static DateFormat medDateForm =
           DateFormat.getDateInstance(DateFormat.MEDIUM, Locale.US);
     static DateFormat longDateForm =
           DateFormat.getDateInstance(DateFormat.LONG, Locale.US);
     static DateFormat fullDateForm =
           DateFormat.getDateInstance(DateFormat.FULL, Locale.US);

    public static void main(String[] args) {
        // Get the current date.
        Date date = new Date();

        // Create string buffer to hold date.
        StringBuffer strbuf = new StringBuffer("");

        // Format the date in short form.
        shortDateForm.format(date, strbuf, dayPos);
        print("\nShort Date: " + strbuf);

        printDayValue(strbuf);
        strbuf.setLength(0);

        shortDateForm.format(date, strbuf, monthPos);
        printMonthValue(strbuf);
        strbuf.setLength(0);

        shortDateForm.format(date, strbuf, yearPos);
        printYearValue(strbuf);
        strbuf.setLength(0);


        // Format the date in medium form
        medDateForm.format(date, strbuf, dayPos);

        print("\nMedium Date: " + strbuf);
        printDayValue(strbuf);
        strbuf.setLength(0);

        medDateForm.format(date, strbuf, monthPos);
        printMonthValue(strbuf);
        strbuf.setLength(0);

        medDateForm.format(date, strbuf, yearPos);
        printYearValue(strbuf);
        strbuf.setLength(0);

        // Format the date in long form
        longDateForm.format(date, strbuf, dayPos);
        print("\nLong Date: " + strbuf);

        printDayValue(strbuf);
        strbuf.setLength(0);

        longDateForm.format(date, strbuf, monthPos);
        printMonthValue(strbuf);
        strbuf.setLength(0);

        longDateForm.format(date, strbuf, yearPos);
        printYearValue(strbuf);
        strbuf.setLength(0);


        // Format the date in full form
        fullDateForm.format(date, strbuf, dayPos);
        print("\nFull Date: " + strbuf);

        printDayValue(strbuf);
        strbuf.setLength(0);

        fullDateForm.format(date, strbuf, monthPos);
        printMonthValue(strbuf);
        strbuf.setLength(0);

        fullDateForm.format(date, strbuf, yearPos);
        printYearValue(strbuf);
        strbuf.setLength(0);

        print("\nFields for Day, Month, Year: ");
        System.out.println(dayPos.getField());
        System.out.println(monthPos.getField());
        System.out.println(yearPos.getField());

    }

    // Bug in JDK: dayPos does not print out (beginIndex and endIndex are 0)
    static void printDayValue(StringBuffer strbuf) {
        System.out.print("Day: ");
        System.out.print(strbuf.toString().substring(dayPos.getBeginIndex(),
                                                     dayPos.getEndIndex()));
        System.out.println(" (" + dayPos.getBeginIndex() + "," +
                                                  dayPos.getEndIndex() + ")");
    }

    static void printMonthValue(StringBuffer strbuf) {
        System.out.print("Month: ");
        print(strbuf.toString().substring(monthPos.getBeginIndex(),
                                          monthPos.getEndIndex()));
    }

    static void printYearValue(StringBuffer strbuf) {
        System.out.print("Year: ");
        print(strbuf.toString().substring(yearPos.getBeginIndex(),
                                                       yearPos.getEndIndex()));
    }

    static void print(String s) {
        System.out.println(s);
    }
}