This is an excellent question but we know very little about such conditions. As Boris Bukh remarked the issue is about points in special positions, because for points in sufficiently general position, even the affine hulls of parts for every partition to r parts will have an empty intersection. However, configurations of points in special positions are of great interest in combinatorial geometry.

One conjecture is that if you have a set of points so that the dimension of (r-1)-Tverberg points is below what can be expected in the generic case then there is a nonempty Tverberg partition to r parts. There are various stengthening and weakening of this conjecture. It was proved by Kadari for the plane.