We consider the following question: given a set of matriceswith no rank-one connections, does it support a nontrivial Young measure limit of gradients? Our main results are these: (a) a Young measure can be supported on four incompatible matrices; (b) in two space dimensions, a Young measure cannot be supported on finitely many incompatible elastic wells; (c) in three or more space dimensions, a Young measure can be supported on three incompatible elastic wells; and (d) ifsupports a nontrivial Young measure with mean value 0, then the linear span ofmust contain a matrix of rank one.