Brain Teaser for the Morning

This one is a hard problem I'm working on at THE BUREAUCRACY, but I'm sure that those with a degree in higher math will be able to assist without a problem:

I have 16 players, 1-16, that are playing each other over the course of 5 weeks. Players will be lumped into groups of four every week to play. The next week, the players are to be put into new groups with players they have never played before, and so on for the 5 weeks.

For example:
Week 1: Group A (1,2,3,4), Group B (5,6,7,8), Group C (9,10,11,12), Group D (13,14,15,16);
Week 2: Group A (1,5,9,13), Group B (2,6,10,14), Group C (3,7,11,15), Group D (4,8,12,16);
Etc. for five weeks.

For Week 3, under the above scenario, 1 could play 6,7,8,10,11,12,14,15,or 16, but if it played 6, it could not play 7,8,10 or 14 (as they've already played 6). And if 1,6, and 11 played, they could not choose 12 or 15 for their fourth (as 11 played 12 in Week 1 and 15 in Week 2), leaving 16. So Group A could be (1,6,11,16), and these players become unavailable for Group B to choose.

It was immediately apparent that 5 weeks is not going to be possible (not enough players), so we'll parce it down to 4 weeks.

We're looking for four weeks of combinations where 16 different players never play against someone they've already played against... or we're looking for a reason why this is impossible.

So if you have time today, and an inkling, give it a shot and stretch that brain.