Transport Layer Security (TLS) is a cornerstone to secure Internet communications. It requires proper deployment and validation of certificate chains. During validation, clients must first construct the chain from server-provided certificates. However, existing research often integrates chain construction into the broader validation process, lacking independent analysis of this crucial step. This paper presents the first systematic assessment of certificate chain construction, covering server-side deployment compliance and client-side capabilities. On the server side, we summarized structural requirements from RFC standards and evaluated real-world website compliance. We found that approximately 3% of Tranco Top 1M domains have deployed non-compliant chains, with common issues including reversed sequences and incomplete chains. The compliance would be influenced by HTTP server and Certificate Authority checks and guidance during the configuration process. On the client side, we evaluated 9 types of chain-building capabilities across 8 mainstream TLS implementations, uncovering prevalent deficiencies like inadequate backtracking and difficulties with long chains. These deficiencies could compromise TLS security, causing a fallback to insecure HTTP or making the service unavailable. Our findings highlight critical gaps in current certificate chain practices. Based on our findings, we also propose recommendations for improving the deployment and construction of certificate chains.