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 suboptimal 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 successive 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 behavior. Comprehensive demonstrations on mainstream MRI reconstruction and synthesis tasks show that ProvoGAN yields superior performance to state-of-the-art volumetric and cross-sectional models.