As large language models continue to scale up, distributed training systems have expanded beyond 10k nodes, intensifying the importance of fault tolerance. Checkpoint has emerged as the predominant fault tolerance strategy, with extensive studies dedicated to optimizing its efficiency. However, the advent of the sparse Mixture-of-Experts (MoE) model presents new challenges due to the substantial increase in model size, despite comparable computational demands to dense models. In this work, we propose the Mixture-of-Checkpoint System (MoC-System) to orchestrate the vast array of checkpoint shards produced in distributed training systems. MoC-System features a novel Partial Experts Checkpointing (PEC) mechanism, an algorithm-system co-design that strategically saves a selected subset of experts, effectively reducing the MoE checkpoint size to levels comparable with dense models. Incorporating hybrid parallel strategies, MoC-System involves fully sharded checkpointing strategies to evenly distribute the workload across distributed ranks. Furthermore, MoC-System introduces a two-level checkpointing management method that asynchronously handles in-memory snapshots and persistence processes. We build MoC-System upon the Megatron-DeepSpeed framework, achieving up to a 98.9% reduction in overhead for each checkpointing process compared to the original method, during MoE model training with ZeRO-2 data parallelism and expert parallelism. Additionally, extensive empirical analyses substantiate that our methods enhance efficiency while maintaining comparable model accuracy, even achieving an average accuracy increase of 1.08% on downstream tasks.