We report on a comprehensive experimental and theoretical study of the quasi-one-dimensional quantum magnet CuNCN. Based on magnetization measurements above room temperature, as well as muon spin rotation and electron spin resonance measurements, we unequivocally establish the localized Cu⁺²-based magnetism and the magnetic transition around 70 K, both controversially discussed in the previous literature. Thermodynamic data conform to the uniform-spin-chain model with a nearest-neighbor intrachain coupling of about 2300 K, in remarkable agreement with the microscopic magnetic model based on density functional theory band-structure calculations. Using exact diagonalization and the coupled-cluster method we derive a collinear antiferromagnetic order with a strongly reduced ordered moment of about 0.4 μ_B, indicating strong quantum fluctuations inherent to this quasi-one-dimensional spin system. We reanalyze the available neutron-scattering data, and conclude that they are not sufficient to resolve or disprove the magnetic order in CuNCN. By contrast, spectroscopic techniques indeed show signatures of long-range magnetic order below 70 K, yet with a rather broad distribution of internal field probed by implanted muons. We contemplate the possible structural origin of this effect and emphasize the strong anisotropic reflection broadening observed with synchrotron powder x-ray diffraction on stoichiometric samples of CuNCN, as well as possible deviations from the ideal CuNCN stoichiometry.