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Perovskite nanoplatelets possess extremely narrow absorption and emission linewidths, which are crucial characteristics for many optical applications. However, their underlying intrinsic and extrinsic line-broadening mechanisms are poorly understood. Here, we apply multi-dimensional coherent spectroscopy to determine the homogeneous line-broadening of colloidal perovskite nanoplatelet ensembles. We demonstrate control of not only their intrinsic linewidths, but also control of various broadening mechanisms by tuning the platelet geometry. Remarkably, we find that decreasing nanoplatelet thickness by a single polyhedral layer results in a 2-fold reduction of the inhomogeneous linewidth and a 3-fold reduction of the intrinsic homogeneous linewidth to the sub-meV regime. In addition, our measurements suggest homogeneously broadened exciton resonances in 3-layer (but not necessarily 4-layer) nanoplatelets at room-temperature.