Convolutional Kolmogorov-Arnold networks (CKANs) extend spline-based Kolmogorov-Arnold networks to convolutional architectures, offering enhanced expressivity and parameter efficiency compared to standard CNNs. However, existing CKAN approaches rely on fixed spline resolutions, leading to training inefficiencies and sensitivity to discretization. We address this gap by introducing a multilevel algorithmic framework for CKANs based on nested spline refinement and exact prolongation. We provide a theoretical interpretation of CKANs as multichannel convolutional neural networks induced by masking and partition-of-unity decompositions of the activation value space. Hierarchical spline refinement and consistent transfer of optimizer state across refinement levels enables true nested iteration. Empirically, refinement accelerates validation accuracy and improves peak performance in image classification tasks, with refinement of deeper spline layers yielding the largest gains. Preliminary neural-operator experiments for scientific machine learning indicate reduced overfitting relative to state-of-the-art architectures, highlighting the potential of multilevel CKANs for hierarchical SciML applications.