Postgres 11

By using db<>fiddle, you agree to license everything you submit by Creative Commons CC0.
CREATE TABLE ProjectCreationTasks ( 
Id                  text NOT NULL PRIMARY KEY,
ProjectName         text,
ProjectCode         text,
DenialReason        text,
CONSTRAINT my_constraint CHECK
 ( (ProjectName IS NULL AND ProjectCode  IS NULL AND DenialReason IS NULL)
   OR ((ProjectName IS NOT NULL AND ProjectCode IS NOT NULL) AND  DenialReason IS NULL)
   OR (DenialReason IS NOT NULL AND ProjectName IS NULL AND ProjectCode  IS NULL)
 )
);

CREATE TABLE


TRUNCATE TABLE ProjectCreationTasks

TRUNCATE TABLE


-- if Name/Code has a value then Reason must be null
INSERT INTO ProjectCreationTasks 
VALUES (1, '1', NULL, NULL)

ERROR:  new row for relation "projectcreationtasks" violates check constraint "my_constraint"
DETAIL:  Failing row contains (1, 1, null, null).


--  Name/Code/Reason can all be null simultaneously, 
INSERT INTO ProjectCreationTasks 
VALUES (2, NULL, NULL, NULL)

INSERT 0 1


-- if Reason has a value then Name/Code must be null 
INSERT INTO ProjectCreationTasks 
VALUES (3, NULL, NULL, 1)

INSERT 0 1


INSERT INTO ProjectCreationTasks 
VALUES (4, 1, NULL, 1)

ERROR:  new row for relation "projectcreationtasks" violates check constraint "my_constraint"
DETAIL:  Failing row contains (4, 1, null, 1).