Graph neural networks (GNNs) are one of the rapidly growing fields within deep learning. While many distributed GNN training frameworks have been proposed to increase the training throughput, they face three limitations when applied to multi-server clusters. 1) They suffer from an inter-server communication bottleneck because they do not consider the inter-/intra-server bandwidth gap, a representative characteristic of multi-server clusters. 2) Redundant memory usage and computation hinder the scalability of the distributed frameworks. 3) Sampling methods, de facto standard in mini-batch training, incur unnecessary errors in multi-server clusters. We found that these limitations can be addressed by exploiting the characteristics of multi-server clusters. Here, we propose GraNNDis, a fast distributed GNN training framework for multi-server clusters. Firstly, we present Flexible Preloading, which preloads the essential vertex dependencies server-wise to reduce the low-bandwidth inter-server communications. Secondly, we introduce Cooperative Batching, which enables memory-efficient, less redundant mini-batch training by utilizing high-bandwidth intra-server communications. Thirdly, we propose Expansion-aware Sampling, a cluster-aware sampling method, which samples the edges that affect the system speedup. As sampling the intra-server dependencies does not contribute much to the speedup as they are communicated through fast intra-server links, it only targets a server boundary to be sampled. Lastly, we introduce One-Hop Graph Masking, a computation and communication structure to realize the above methods in multi-server environments. We evaluated GraNNDis on multi-server clusters, and it provided significant speedup over the state-of-the-art distributed GNN training frameworks. GraNNDis is open-sourced at https://github.com/AIS-SNU/GraNNDis_Artifact to facilitate its use.