Puiseux series is a mathematical tool for dynamics analysis of time-delay systems near a repeated characteristic root, and it has became a hot research topic in the recent years. This paper presents a new observation that under certain conditions, contradictions occur in calculating the Puiseux series expansion of time-delay systems when the method of undetermined coefficients is used, the reason that leads to contradictions, as well as some normal forms of the correct Puiseux series expansion under some specified conditions, are given.