Magnetic resonance imaging (MRI) offers the flexibility to image a given anatomic volume under a multitude of tissue contrasts. Yet, scan time considerations put stringent limits on the quality and diversity of MRI data. The gold-standard approach to alleviate this limitation is to recover high-quality images from data undersampled across various dimensions, most commonly the Fourier domain or contrast sets. A primary distinction among recovery methods is whether the anatomy is processed per volume or per cross-section. Volumetric models offer enhanced capture of global contextual information, but they can suffer from sub-optimal learning due to elevated model complexity. Cross-sectional models with lower complexity offer improved learning behavior, yet they ignore contextual information across the longitudinal dimension of the volume. Here, we introduce a novel progressive volumetrization strategy for generative models (ProvoGAN) that serially decomposes complex volumetric image recovery tasks into succes-sive cross-sectional mappings task-optimally ordered across individual rectilinear dimensions. ProvoGAN effectively captures global context and recovers fine-structural details across all dimensions, while maintaining low model complexity and improved learning behaviour. Comprehensive demonstrations on mainstream MRI reconstruction and synthesis tasks show that ProvoGAN yields superior per-formance to state-of-the-art volumetric and cross-sectional models.