While previous studies have explored attacks via random, simple, and skilled forgeries, generative attacks have received limited attention in the data-driven signature verification (DASV) process. Thus, this paper explores the impact of generative attacks on DASV and proposes practical and interpretable countermeasures. We investigate the power of two prominent Deep Generative Models (DGMs), Variational Auto-encoders (VAE) and Conditional Generative Adversarial Networks (CGAN), on their ability to generate signatures that would successfully deceive DASV. Additionally, we evaluate the quality of generated images using the Structural Similarity Index measure (SSIM) and use the same to explain the attack's success. Finally, we propose countermeasures that effectively reduce the impact of deep generative attacks on DASV. We first generated six synthetic datasets from three benchmark offline-signature datasets viz. CEDAR, BHSig260- Bengali, and BHSig260-Hindi using VAE and CGAN. Then, we built baseline DASVs using Xception, ResNet152V2, and DenseNet201. These DASVs achieved average (over the three datasets) False Accept Rates (FARs) of 2.55%, 3.17%, and 1.06%, respectively. Then, we attacked these baselines using the synthetic datasets. The VAE-generated signatures increased average FARs to 10.4%, 10.1%, and 7.5%, while CGAN-generated signatures to 32.5%, 30%, and 26.1%. The variation in the effectiveness of attack for VAE and CGAN was investigated further and explained by a strong (rho = -0.86) negative correlation between FARs and SSIMs. We created another set of synthetic datasets and used the same to retrain the DASVs. The retained baseline showed significant robustness to random, skilled, and generative attacks as the FARs shrank to less than 1% on average. The findings underscore the importance of studying generative attacks and potential countermeasures for DASV.