The easiest way to understand this to do the operation in two steps.

First create a view which returns each distinct case_number together with
its number of payment_id's.

CREATE VIEW caseview (case_number, number_of_payment_ids) AS
 SELECT case_number, COUNT(*)
 FROM casetable
 GROUP BY case_number

This view will return the following:
SQL>select * from caseview;
CASE_NUMBER NUMBER_OF_PAYMENT_IDS
=========== =====================
      1000                     3
      1001                     1
      1002                     1
      1003                     3
      1004                     1

You can now use the view to get the wanted result:

SELECT number_of_payment_ids,
      COUNT(*) AS number_of_case_numbers
FROM caseview
GROUP BY number_of_payment_ids;

HTH,
Jarl

How about this?
SELECT payment_cnt AS "Number of payment ids"
    , COUNT(*)    AS "Number of case numbers"
 FROM (SELECT COUNT(payment_id) AS payment_cnt
         FROM case_payment
        GROUP BY
              case_number
      ) AS Q
GROUP BY
      payment_cnt