Graph Convolutional Networks (GCNs) have become a standard approach for semi-supervised node classification, yet practitioners lack clear guidance on when GCNs provide meaningful improvements over simpler baselines. We present a diagnostic study using the Amazon Computers co-purchase data to understand when and why GCNs help. Through systematic experiments with simulated label scarcity, feature ablation, and per-class analysis, we find that GCN performance depends critically on the interaction between graph homophily and feature quality. GCNs provide the largest gains under extreme label scarcity, where they leverage neighborhood structure to compensate for limited supervision. Surprisingly, GCNs can match their original performance even when node features are replaced with random noise, suggesting that structure alone carries sufficient signal on highly homophilous graphs. However, GCNs hurt performance when homophily is low and features are already strong, as noisy neighbors corrupt good predictions. Our quadrant analysis reveals that GCNs help in three of four conditions and only hurt when low homophily meets strong features. These findings offer practical guidance for practitioners deciding whether to adopt graph-based methods.