Generative Adversarial Networks (GAN) have many potential medical imaging applications, including data augmentation, domain adaptation, and model explanation. Due to the limited embedded memory of Graphical Processing Units (GPUs), most current 3D GAN models are trained on low-resolution medical images. In this work, we propose a novel end-to-end GAN architecture that can generate high-resolution 3D images. We achieve this goal by separating training and inference. During training, we adopt a hierarchical structure that simultaneously generates a low-resolution version of the image and a randomly selected sub-volume of the high-resolution image. The hierarchical design has two advantages: First, the memory demand for training on high-resolution images is amortized among subvolumes. Furthermore, anchoring the high-resolution subvolumes to a single low-resolution image ensures anatomical consistency between subvolumes. During inference, our model can directly generate full high-resolution images. Experiments on 3D thorax CT and brain MRI demonstrate that our approach outperforms baselines in quality of generated images.