Concept Bottleneck Models (CBMs) enhance the interpretability of AI systems, particularly by bridging visual input with human-understandable concepts, effectively acting as a form of multimodal interpretability model. However, existing CBMs typically assume static datasets, which fundamentally limits their adaptability to real-world, continuously evolving multimodal data streams. To address this, we define a novel continual learning task for CBMs: simultaneously handling concept-incremental and class-incremental learning. This task requires models to continuously acquire new concepts (often representing cross-modal attributes) and classes while robustly preserving previously learned knowledge. To tackle this challenging problem, we propose CONceptual Continual Incremental Learning (CONCIL), a novel framework that fundamentally re-imagines concept and decision layer updates as linear regression problems. This reformulation eliminates the need for gradient-based optimization, thereby effectively preventing catastrophic forgetting. Crucially, CONCIL relies solely on recursive matrix operations, rendering it highly computationally efficient and well-suited for real-time and large-scale multimodal data applications. Experimental results compellingly demonstrate that CONCIL achieves "absolute knowledge memory" and significantly surpasses the performance of traditional CBM methods in both concept- and class-incremental settings, thus establishing a new paradigm for continual learning in CBMs, particularly valuable for dynamic multimodal understanding.