We consider the head-on collision of equal-mass neutron stars boosted towards each other and we study the behaviour of such systems near the threshold of black-hole formation. In particular, we confirm the previous findings by Jin K J et al (2007 Phys. Rev. Lett 98 131101) that a type I critical phenomenon can be observed by fine-tuning the initial mass of the two neutron stars. At the same time, we argue against the interpretation that the critical solution is not a perturbed spherical star and show instead that the metastable star corresponds to a (perturbed) equilibrium solution on the unstable branch of the equilibrium configurations. As a result, the head-on collision of two neutron stars near the critical threshold can be seen as a transition in the space of configurations from an initial stable solution over to a critical metastable one which can either migrate to a stable solution or collapse to a black hole. The critical exponent for this process shows a fine structure which was already observed in the case of the critical collapse of scalar fields but never before for perfect fluids.