I must be misunderstanding something. Let's look at the following two definitions for a set of points $S=\{v_1,v_2,...,v_k\}$ to be affinely independent:
1) S is affinely independent if the set $\{v_2-v_1, v_3-v_1, ..., v_k-v_1 \}$ is linearly independent. see http://homepages.rpi.edu/~mitchj/handouts/faces/.
2) If no three points in $S$ lie on a line, no 4 points lie on a plane, no 5 points lie on a 3-dimensional subspace, etc. See Affine dimension of a simplex
Here is an example to show that the two definitions are inequivalent: take one facet of a cube. If we translate it such that it includes the origin, we get that it has affine dimension 2 according to definition 1.
However, if we take three vertices from that facet, we find that they do not lie on a line, and therefore they are affinely independent. Hence the affine dimension of the facet is 3.
Thanks.