We report experimental and theoretical studies on the magnetic and thermodynamic properties of NIT-2Py, a free radical based organic magnet. From magnetization and specific-heat measurements we establish the temperature versus magnetic field phase diagram which includes two Bose-Einstein condensates (BEC) and an infrequent half-magnetization plateau. Calculations based on density functional theory demonstrate that magnetically this system can be mapped to a quasi-two-dimensional structure of weakly coupled tetramers. Density matrix renormalization group calculations show the unusual characteristics of the BECs where the spins forming the low-field condensate are different than those participating in the high-field one.