Phenalenyl, an open-shell neutral radical that can form both π-stacked dimers and conducting molecular crystals, has gained attention for its interesting and potentially useful electrical and magnetic properties. The properties of this complex physical system are fairly well understood, making it an ideal testing ground for the newly developed van der Waals density functional (vdW-DF). We invoke a simple approximation, allowing the vdW-DF to be used within spin-polarized density functional theory and test this approximation on the π-stacked phenalenyl dimer. The results indicate that the vdW-DF is capable of qualitatively describing the interaction between two neutral radicals in the π-stacked configuration, producing, in line with experiment, binding distances that are significantly below the sum of the van der Waals radii. This is a nontypical distance range where most other theories fail. We then investigate two hypothetical closed-shell analogues of this dimer, one formed by replacing the central carbon of phenalenyl with a nitrogen atom and the other formed by replacing the central carbon with a boron atom. In these cases, relatively strong interaction energies are obtained at more typical equilibrium distances for van der Waals dimers. The nitrogen-substituted dimer shows an unexpected rotational barrier that is dictated by the electronic kinetic energy within the system. The torsional curve of the boron-substituted dimer also exhibits a rotational barrier, but this is found to disappear when exact exchange is used in place of a local or semilocal exchange functional.