We have investigated the local atomic structures of several compositions of the amorphous phase of the system CuxGe50-xTe50 (0 ≤ x ≤ 33.3), based on anomalous X-ray scattering (AXS) as well as extended X-ray absorption fine-structure (EXAFS) experiments, and discuss the unusual trend regarding their thermal stability as a function of the Cu content. At low concentrations, Cu tends to agglomerate in nanoclusters reminiscent of the crystalline phase of metallic Cu, leading to a more and more Ge-deficient Ge-Te host network structure with growing Cu content and an increasing thermal stability. At higher Cu concentrations, Cu is incorporated into the network, leading to an overall weaker bonding situation which is associated with a decreasing thermal stability.