Hi Derek,

Yes, it is possible. It will take a second to explain.

There is a schedule for 16 players to play 5 rounds in groups of 4 so that each pair of players is together exactly once. Copied below is the 5-round schedule you can find here: mathcentral.uregina.ca/QQ/database/QQ.09.08/h/bill4.html. The one you need is obtained by modifying it just a bit.

day 1: {1,2,3,4} {5,6,7,8} {9,a,b,c} {d,e,f,g}

day 2: {1,5,9,d} {2,6,a,e} {3,7,b,f} {4,8,c,g}

day 3: {1,6,c,f} {2,5,b,g} {3,8,a,d} {4,7,9,e}

day 4: {1,b,8,e} {2,c,7,d} {3,9,6,g} {4,a,5,f}

day 5: {1,7,a,g} {2,8,9,f} {3,5,c,e} {4,b,6,d}

If you let the players in group A be 1,2,3,4,5,6,7,8 and the ones in group B be group 9,a,b,c,d,e,f,g, then your schedule is obtained by deleting day 1 from the schedule above (that is, your 4 days are numbered 2, 3, 4, 5 above).

Enjoy!
--Victoria