Single cells often generate precise responses by involving dissipative out-of-thermodynamic equilibrium processes in signaling networks. The available free energy to fuel these processes could become limited depending on the metabolic state of an individual cell. How does limiting dissipation affect the kinetics of high precision responses in single cells? I address this question in the context of a kinetic proofreading scheme used in a simple model of early time T cell signaling. I show using exact analytical calculations and numerical simulations that limiting dissipation qualitatively changes the kinetics in single cells marked by emergence of slow kinetics, large cell-to-cell variations of copy numbers, temporally correlated stochastic events (dynamic facilitation), and, ergodicity breaking. Thus, constraints in energy dissipation, in addition to negatively affecting ligand discrimination in T cells, can create a fundamental difficulty in interpreting single cell kinetics from cell population level results.