It is possible to play 5 rounds of foursomes where every players plays with 15 different opponents.
(A B C D) (E F G H) (I J K L) (M N O P) (Q R S T)
(A E I M) (B J O Q) (C H N T) (D G L S) (F K P R)
(A G K O) (B I P T) (C F M S) (D H J R) (E L N Q)
(A H L P) (B K N S) (C E O R) (D F I Q) (G J M T)
(A F J N) (B L M R) (C G P Q) (D E K T) (H I O S)
However, I think a 6th round will allways introduce repeated pairings and leave other pairings unplayed. Perhaps you could try something different for the final day? If you were keeping score, then you could use the rankings after 5 days to select the foursomes for the final day.
Adding a 6th day the schedule above is not the best thing to do. I think we need to start again.
Round 1
16 13 2 14
17 8 18 1
9 10 5 3
15 20 4 6
7 19 12 11
Round 2
10 13 17 11
7 14 1 15
4 3 19 16
12 18 9 20
2 5 8 6
Round 3
15 19 8 10
6 14 11 9
12 2 17 4
13 16 7 1
3 5 20 18
Round 4
14 4 18 10
1 9 2 19
17 3 6 7
5 12 15 13
20 11 16 8
Round 5
20 7 10 2
8 12 14 3
16 15 9 17
11 1 5 4
18 6 13 19
Round 6
9 4 13 8
19 17 14 20
18 7 16 5
1 10 6 12
11 2 3 15
Above there are 16 pairs that don't occur together and 6 pairs that occur twice. The latter are (1,7) (3,5) (5,18) (7,16) (13,16) (18,20).