Deep cerebellar nuclei are a key structure of the cerebellum that are involved in processing motor and sensory information. It is thus a crucial step to accurately segment deep cerebellar nuclei for the understanding of the cerebellum system and its utility in deep brain stimulation treatment. However, it is challenging to clearly visualize such small nuclei under standard clinical magnetic resonance imaging (MRI) protocols and therefore precise segmentation is not feasible. Recent advances in 7 Tesla (T) MRI technology and great potential of deep neural networks facilitate automatic patient-specific segmentation. In this paper, we propose a novel deep learning framework (referred to as DCN-Net) for fast, accurate, and robust patient-specific segmentation of deep cerebellar dentate and interposed nuclei on 7T diffusion MRI. DCN-Net effectively encodes contextual information on the patch images without consecutive pooling operations and adding complexity via proposed dilated dense blocks. During the end-to-end training, label probabilities of dentate and interposed nuclei are independently learned with a hybrid loss, handling highly imbalanced data. Finally, we utilize self-training strategies to cope with the problem of limited labeled data. To this end, auxiliary dentate and interposed nuclei labels are created on unlabeled data by using DCN-Net trained on manual labels. We validate the proposed framework using 7T B0 MRIs from 60 subjects. Experimental results demonstrate that DCN-Net provides better segmentation than atlas-based deep cerebellar nuclei segmentation tools and other state-of-the-art deep neural networks in terms of accuracy and consistency. We further prove the effectiveness of the proposed components within DCN-Net in dentate and interposed nuclei segmentation.