Diffuse interstellar bands (DIBs) are mysterious absorption lines in the optical spectra of stars, and have been known for 75 years. Although it is widely believed that they arise from gas-phase organic molecules (rather than from dust grains) in the interstellar medium, no consensus has been reached regarding their precise cause. The realization that many emission features in astronomical infrared spectra probably arise from polycyclic aromatic hydrocarbons (PAHs), which may themselves be very abundant in the interstellar medium, has led to the suggestion that ionized PAHs might be the source of the DIBs. Laboratory investigations have revealed that small, positively charged PAHs in matrices have absorption features that bear some resemblance to DIBs, but no clear identification of any DIB with any specific PAH cation has yet been made. Here we report a laboratory study of the chemical reactivity of PAH cations (C6H6+, C10H8+ and C16H10+) in the gas phase. We find that these PAH cations are very reactive, and are therefore unlikely to survive in high abundances in the interstellar medium. Rather, such molecules will react rapidly with hydrogen, and we therefore suggest that the resulting protonated PAH cations (and species derived from them) should become the focus of future searches for a correspondence between molecular absorption features and the DIBs.