You might have duplicate rows too. You could use Analytic ranking function to fetch the required rows.
Set up:
SQL> CREATE TABLE Table1
  2      (PERSON_ID int, EFFECTIVE_END_DATE varchar2(21), ASSIGNMENT_ID int, FULL_NAME varchar2(7));
Table created.
SQL>
SQL> INSERT ALL
  2      INTO Table1 (PERSON_ID, EFFECTIVE_END_DATE, ASSIGNMENT_ID, FULL_NAME)
  3           VALUES (33151, to_date('2013-08-04','YYYY-MM-DD'), 33885, 'Test, C')
  4      INTO Table1 (PERSON_ID, EFFECTIVE_END_DATE, ASSIGNMENT_ID, FULL_NAME)
  5           VALUES (33151, to_date('2013-10-04','YYYY-MM-DD'), 33885, 'Test, C')
  6      INTO Table1 (PERSON_ID, EFFECTIVE_END_DATE, ASSIGNMENT_ID, FULL_NAME)
  7           VALUES (33151, to_date('2015-02-19','YYYY-MM-DD'), 33885, 'Test, C')
  8      INTO Table1 (PERSON_ID, EFFECTIVE_END_DATE, ASSIGNMENT_ID, FULL_NAME)
  9           VALUES (33151, to_date('2013-08-04','YYYY-MM-DD'), 33885, 'Test, C')
 10      INTO Table1 (PERSON_ID, EFFECTIVE_END_DATE, ASSIGNMENT_ID, FULL_NAME)
 11           VALUES (33151, to_date('2013-10-04','YYYY-MM-DD'), 33885, 'Test, C')
 12      INTO Table1 (PERSON_ID, EFFECTIVE_END_DATE, ASSIGNMENT_ID, FULL_NAME)
 13           VALUES (33151, to_date('2015-02-19','YYYY-MM-DD'), 33885, 'Test, C')
 14  SELECT * FROM dual;
6 rows created.
SQL>
SQL> COMMIT;
Commit complete.
SQL>
SQL> SELECT * FROM table1;
 PERSON_ID EFFECTIVE_END_DATE    ASSIGNMENT_ID FULL_NA
---------- --------------------- ------------- -------
     33151 04-AUG-13                     33885 Test, C
     33151 04-OCT-13                     33885 Test, C
     33151 19-FEB-15                     33885 Test, C
     33151 04-AUG-13                     33885 Test, C
     33151 04-OCT-13                     33885 Test, C
     33151 19-FEB-15                     33885 Test, C
6 rows selected.
SQL>
Test case:
SQL> WITH data AS
  2    (SELECT t.*,
  3      rank() over(partition BY person_id order by person_id, EFFECTIVE_END_DATE DESC) rn
  4    FROM table1 t
  5    )
  6  SELECT PERSON_ID,
  7    EFFECTIVE_END_DATE,
  8    ASSIGNMENT_ID,
  9    FULL_NAME
 10  FROM data
 11  WHERE rn = 1;
 PERSON_ID EFFECTIVE_END_DATE    ASSIGNMENT_ID FULL_NA
---------- --------------------- ------------- -------
     33151 19-FEB-15                     33885 Test, C
     33151 19-FEB-15                     33885 Test, C
SQL>