Fact-checking pipelines increasingly adopt the Decompose-Then-Verify paradigm, where texts are broken down into smaller claims for individual verification and subsequently combined for a veracity decision. While decomposition is widely-adopted in such pipelines, its effects on final fact-checking performance remain underexplored. Some studies have reported improvements from decompostition, while others have observed performance declines, indicating its inconsistent impact. To date, no comprehensive analysis has been conducted to understand this variability. To address this gap, we present an in-depth analysis that explicitly examines the impact of decomposition on downstream verification performance. Through error case inspection and experiments, we introduce a categorization of decomposition errors and reveal a trade-off between accuracy gains and the noise introduced through decomposition. Our analysis provides new insights into understanding current system's instability and offers guidance for future studies toward improving claim decomposition in fact-checking pipelines.