Transport Layer Security (TLS) relies on correct deployment, construction, and validation of certificate chains. This paper presents the first systematic assessment of certificate chain construction from both server-side deployment and client-side behavior. By measuring Tranco Top 1M domains, the study finds that approximately 3% deploy non-compliant chains, with reversed sequences and incomplete chains among the most common problems. It further evaluates nine chain-building capabilities across eight mainstream TLS implementations and identifies deficiencies such as inadequate backtracking and difficulties handling long chains. These inconsistencies can lead to TLS validation failures, service unavailability, or fallback to insecure HTTP. The paper concludes with recommendations for improving certificate-chain deployment and construction practices across the Web PKI ecosystem.