We study a system of reflected Brownian motions on the positive half-line in which each particle has a drift toward the origin determined by the local times at the origin of all the particles. If this local time drift is too strong, such systems exhibit a breakdown in their solutions in that there is a time beyond which the system cannot be extended. In the finite particle case we give a complete characterisation of this finite time breakdown, relying on a novel dynamic graph structure. We consider the mean-field limit of the system in the symmetric setting, which admits a McKean--Vlasov representation, and establish propagation of chaos. In the absence of breakdowns, the McKean--Vlasov equation exhibits multiple stationary and unique self-similar solutions and we prove convergence to these profiles. This work is motivated by models for liquidity in financial markets, the supercooled Stefan problem, and a toy model for cell polarisation.