We study whether the signal seen by pulsar timing arrays (PTAs) may originate from gravitational waves (GWs) induced by large primordial perturbations. Such perturbations may be accompanied by a sizable primordial black hole (PBH) abundance. We improve existing analyses and show that PBH overproduction disfavors Gaussian scenarios for scalar-induced GWs at 2σ and single-field inflationary scenarios, accounting for non-Gaussianity, at 3σ as the explanation of the most constraining NANOGrav 15-year data. This tension can be relaxed in models where non-Gaussianities suppress the PBH abundance. On the flip side, the PTA data does not constrain the abundance of PBHs.