Electron attachment properties of covalent molecules and ion clusters with vanishing dipole moments but large quadrupoles are studied with coupled cluster ab initio methods. Selection of the molecules studied is driven by two goals, finding a paradigm quadrupole-bound anion and investigating whether there is a correlation between the magnitude of the molecular quadrupole and the vertical attachment energy. Out of all examined species, only the ion clusters and four of the covalent molecules are found to support bound anions. The shapes and spatial extents of the associated excess electron distributions are qualitatively and quantitatively characterized, respectively. Two of the four covalent systems are especially promising as paradigm systems because of advantageous trade-offs regarding the number of isomers and conformers as well as synthetic closeness to commercial sources. No correlation was found between the vertical attachment energy and molecular quadrupole in an analysis that included the newly identified bound anions, those molecules, which were found not to support bound anions, and succinonitrile, which had been studied before. Moreover, there is clearly no such thing as a "critical quadrupole moment". There are, however, very strong electron correlation effects involved in the binding of the excess electrons, and similar to succinonitrile, for five out of six anions identified here, the molecular quadrupole of the neutral itself is too weak to bind an excess electron, and electron correlation in the form of dynamic polarization is required to do so.