Performance-influence models are beneficial for understanding how configurations affect system performance, but their creation is challenging due to the exponential growth of configuration spaces. While gray-box approaches leverage selective "structural knowledge" (like the module execution graph of the system) to improve modeling, the relationship between this knowledge, a system's characteristics (we call them "structural aspects"), and potential model improvements is not well understood. This paper addresses this gap by formally investigating how variations in structural aspects (e.g., the number of modules and options per module) and the level of structural knowledge impact the creation of "opportunities" for improved "modular performance modeling". We introduce and quantify the concept of modeling "hardness", defined as the inherent difficulty of performance modeling. Through controlled experiments with synthetic system models, we establish an "analytical matrix" to measure these concepts. Our findings show that modeling hardness is primarily driven by the number of modules and configuration options per module. More importantly, we demonstrate that both higher levels of structural knowledge and increased modeling hardness significantly enhance the opportunity for improvement. The impact of these factors varies by performance metric; for ranking accuracy (e.g., in debugging task), structural knowledge is more dominant, while for prediction accuracy (e.g., in resource management task), hardness plays a stronger role. These results provide actionable insights for system designers, guiding them to strategically allocate time and select appropriate modeling approaches based on a system's characteristics and a given task's objectives.