CsPbI3 is a promising material for optoelectronics owing to its thermal robustness and favorable bandgap. However, its fabrication is challenging because its photoactive phase is thermodynamically unstable at room temperature. Adding dimethylammonium (DMA) alleviates this instability and is currently understood to result in the formation of DMA(x)Cs(1-x)PbI(3) perovskite solid solutions. Here, we use NMR of the Cs-133 and C-13 local structural probes to show that these solid solutions are not thermodynamically stable, and their synthesis under thermodynamic control leads to a segregated mixture of yellow one-dimensional DMAPbI(3) phase and delta-CsPbI3. We show that mixed-cation DMA(x)Cs(1-x)PbI(3) perovskite phases only form when they are kinetically trapped by rapid antisolvent-induced crystallization. We explore the energetics of DMA incorporation into CsPbI3 using first-principles calculations and molecular dynamics simulations and find that this process is energetically unfavorable. Our results provide a complete atomic-level picture of the mechanism of DMA-induced stabilization of the black perovskite phase of CsPbI3 and shed new light on this deceptively simple material.