It is indeed possible to have a schedule for 5 rounds:
(1 17 16 14) (4 8 13 3) (11 20 7 15) (12 6 5 18) (2 9 10 19)
(2 18 17 15) (5 9 14 4) (12 16 8 11) (13 7 1 19) (3 10 6 20)
(3 19 18 11) (1 10 15 5) (13 17 9 12) (14 8 2 20) (4 6 7 16)
(4 20 19 12) (2 6 11 1) (14 18 10 13) (15 9 3 16) (5 7 8 17)
(5 16 20 13) (3 7 12 2) (15 19 6 14) (11 10 4 17) (1 8 9 18)
Fortunately you did not ask for a 6 round schedule, on the face of it, this might seem to be possible - 6 partners and 12 opponents, but it has been proven by the mathematicians that it can't exist.
Laurie,
If you only have 12 players then it is only possible to play a single round. No matter how you arrange the second round there will be at least three pairs of players who meet for a second time. With 24 players it is possible to play up to seven rounds and still not have any repeated pairings. For example:
(16 20 21 9) (5 2 3 19) (11 13 10 1) (14 15 7 22) ( 8 17 4 23) (12 6 18 24)
(17 21 15 10) (6 3 4 20) (12 14 11 2) ( 8 16 1 22) ( 9 18 5 23) (13 7 19 24)
(18 15 16 11) (7 4 5 21) (13 8 12 3) ( 9 17 2 22) (10 19 6 23) (14 1 20 24)
(19 16 17 12) (1 5 6 15) (14 9 13 4) (10 18 3 22) (11 20 7 23) ( 8 2 21 24)
(20 17 18 13) (2 6 7 16) ( 8 10 14 5) (11 19 4 22) (12 21 1 23) ( 9 3 15 24)
(21 18 19 14) (3 7 1 17) ( 9 11 8 6) (12 20 5 22) (13 15 2 23) (10 4 16 24)
(15 19 20 8) (4 1 2 18) (10 12 9 7) (13 21 6 22) (14 16 3 23) (11 5 17 24)
So just take any five rounds from the above for your competition.