Given a graph, we ask whether it is possible to find a flexible labelling, namely, edge lengths such that there are infinitely many compatible realizations, modulo rigid motions. Even if a graph is generically rigid, then non-generic edge lengths may still be flexible. A classical construction by A. Dixon from 1899 works for all bipartite graphs. In this talk, we give a combinatorial criterion for the existence of a flexible labelling. This is joint work with Georg Grasegger and Josef Schicho.