The common practice for assessing automatic evaluation metrics is to measure the correlation between their induced system rankings and those obtained by reliable human evaluation, where a higher correlation indicates a better metric. Yet, an intricate setting arises when an NLP task is evaluated by multiple Quality Criteria (QCs), like for text summarization where prominent criteria include relevance, consistency, fluency and coherence. In this paper, we challenge the soundness of this methodology when multiple QCs are involved, concretely for the summarization case. First, we show that the allegedly best metrics for certain QCs actually do not perform well, failing to detect even drastic summary corruptions with respect to the considered QC. To explain this, we show that some of the high correlations obtained in the multi-QC setup are spurious. Finally, we propose a procedure that may help detect this effect. Overall, our findings highlight the need for further investigating metric evaluation methodologies for the multiple-QC case.