Postgres 15

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CREATE TABLE test_data (
  id integer PRIMARY KEY
, message VARCHAR(10)
, datetimestamp timestamp NOT NULL
, server VARCHAR(10)
, system VARCHAR(10)
);

INSERT INTO test_data VALUES
  (09, 'Normal' , '2022-07-13 08:59:09', 'Server 1' , 'System 1')
, (10, 'Normal' , '2022-07-13 08:59:10', 'Server 4' , 'System 2')
, (11, 'Normal' , '2022-07-13 08:59:11', 'Server 3' , 'System 3')
, (12, 'Warning', '2022-07-13 08:59:12', 'Server 35', 'System 27')
, (13, 'Warning', '2022-07-13 08:59:13', 'Server 5' , 'System 5')
, (14, 'Warning', '2022-07-13 08:59:14', 'Server 9' , 'System 6')
, (15, 'Warning', '2022-07-13 08:59:15', 'Server 8' , 'System 7')
, (16, 'Error'  , '2022-07-13 08:59:16', 'Server 12', 'System 8')
, (17, 'Error'  , '2022-07-13 08:59:17', 'Server 15', 'System 9')
, (18, 'Warning', '2022-07-13 08:59:18', 'Server 29', 'System 10')
, (19, 'Warning', '2022-07-13 08:59:19', 'Server 22', 'System 11')
, (20, 'Warning', '2022-07-13 08:59:20', 'Server 13', 'System 12')
, (21, 'Normal' , '2022-07-13 08:59:21', 'Server 16', 'System 13')
, (22, 'Normal' , '2022-07-13 08:59:22', 'Server 19', 'System 14')
, (23, 'Normal' , '2022-07-13 08:59:23', 'Server 21', 'System 15')
, (24, 'Warning', '2022-07-13 08:59:24', 'Server 24', 'System 16')
, (25, 'Warning', '2022-07-13 08:59:25', 'Server 27', 'System 17')
, (26, 'Warning', '2022-07-13 08:59:26', 'Server 25', 'System 18')
, (27, 'Error'  , '2022-07-13 08:59:27', 'Server 30', 'System 23')
, (28, 'Error'  , '2022-07-13 08:59:28', 'Server 31', 'System 20')

, (29, 'Warning', '2022-07-14 08:59:29', 'Server 66', 'System 67')  -- added single-row warning
, (30, 'Error'  , '2022-07-14 08:59:30', 'Server 68', 'System 69')
;

22 rows affected

WITH cte AS (
   SELECT grp, datetimestamp, message, server, system
        , bool_or(message = 'Error' AND last_msg = 'Warning') OVER (PARTITION BY grp) AS report  -- qualifies for report
   FROM  (
      SELECT *
           , count(*) FILTER (WHERE message = 'Warning' AND last_msg <> 'Warning') OVER (ORDER BY datetimestamp) AS grp
      FROM  (
         SELECT datetimestamp, message, server, system
              , lag(message) OVER (ORDER BY datetimestamp) AS last_msg
         FROM   test_data
         ) sub1
      WHERE  message IN ('Warning', 'Error')  -- trim noise early
      ) sub2
   )
SELECT a.datetimestamp AS start_time, z.datetimestamp AS end_time
     , a.server AS start_server, z.server AS end_server
     , a.system AS start_system, z.system AS end_system
FROM  (
   SELECT DISTINCT ON (grp) *
   FROM   cte
   WHERE  report
   ORDER  BY grp, datetimestamp
   ) a
JOIN  (
   SELECT DISTINCT ON (grp) *
   FROM   cte
   WHERE  report
   AND    message = 'Warning'  -- exclude trailing errors
   ORDER  BY grp, datetimestamp DESC
   ) z USING (grp);

start_time	end_time	start_server	end_server	start_system	end_system
2022-07-13 08:59:12	2022-07-13 08:59:15	Server 35	Server 8	System 27	System 7
2022-07-13 08:59:24	2022-07-13 08:59:26	Server 24	Server 25	System 16	System 18
2022-07-14 08:59:29	2022-07-14 08:59:29	Server 66	Server 66	System 67	System 67