The strong deformation of a thin sheet or shell is characterized by the formation of sharp structures. In a shell that is forced to collapse, these features develop folds that make the process more efficient. On the other hand, in the crumpling of a membrane, the same features tend to stiffen the structure. We experimentally investigated a mechanism relevant in the collapse of shells and the early stages of crumpling, namely the mutual annihilation of vertex singularities. We show that the annihilation of two vertices in a displacement-controlled experiment is accompanied by a decrease of the forces at equilibrium. We propose a simple theoretical interpretation for this instability based on the near inextensibility of thin elastic sheets.