ORA-00001 - But no name
A possible bug for not displaying the constraint name with bulk DML and SAVE EXCEPTIONS clause:

Bug 7210333 : CONSTRAINT NAME IN SQLERRM IS NOT OUTPUT WHEN SAVE EXCEPTIONS IS USED

To catch the records causing the error, you could use DML error logging as well:

create table t1(c1 number unique);

begin
  dbms_errlog.create_error_log('T1', 'T1_ERRORS');
end;
/


Insert data:

declare
  type t_varray is varray(4) of number;
  l_varray t_varray := t_varray(1, 2, 2, 3);
begin
  forall i in 1..l_varray.count
  insert into t1 values (l_varray(i)) 
    log errors into t1_errors
    reject limit unlimited;
end;
/


Actual data:

select * from t1;

        C1
----------
         1
         2
         3


Rows violating the constraint with row data and constraint name:

select * from t1_errors;

ORA_ERR_NUMBER$ ORA_ERR_MESG$                                                ORA_ERR_ROWID$       ORA_ERR_OPTYP$       ORA_ERR_TAG$         C1
--------------- ------------------------------------------------------------ -------------------- -------------------- -------------------- ----------
              1 ORA-00001: unique constraint (BP.SYS_C0013693) violated                           I                                         2