Ligand-receptor binding is a key ingredient of T-cell signaling, wherein extracellular ligands bind T-cell receptors (TCRs) to initiate an intracellular cascade. This signaling requires T-cells to recognize different ligands, and map those ligands to different intracellular functions. Puzzlingly, many cellular signaling pathways, including the TCR signaling pathway, have a bottleneck shape, meaning that many different ligands bind to the same receptor. It remains unclear how cells overcome this bottleneck to produce coherent and distinct signals in the cell interior, based on external ligand identities and concentrations. We posit that T-cells navigate this bottleneck by classifying sets of extracellular ligand identities and concentrations into discrete classes of intracellular responses, rather than having every concentration and type of extracellular ligand lead to a unique intracellular response. We establish the ability of a TCR pathway to behave as a classifier by approximating a TCR signaling pathway using an architecturally similar neural network. We show that such a neural network is indeed able to classify extracellular ligand concentrations into distinct intracellular responses. We computationally model a true TCR system, and show that the behavior of the true system is qualitatively similar to that of the TCR-like neural network. Finally, we show that overcoming this bottleneck is not possible for all receptor types, but is instead unique to certain types of receptors. In conclusion, our results show that TCRs have the capacity to act as classification machines to sort extracellular ligands into biologically distinct responses.