What do you mean by "even grid of 2x2"? Off-hand, I would say that since you want every team to play every other team in every event, the fact that there are multiple events does not complicate the problem, since each event is just going to be a copy of the first event. So examine one event. Every team playing every other team exactly once sounds an awful lot like a complete graph to me. The vertices are teams, and the edges are games. If there are n teams (16 in your case), there will be n(n-1)/2 games played. So in your case, each event will have 120 games played. Multiply by 8 events and you get a whopping 960 games!
So I would say there's a solution no matter how many teams you have. Or am I missing something here?