Legendre polynomial method is well-known in modeling acoustic wave characteristics. This method uses for the mechanical displacements a single polynomial expansion over the entire sandwich layers. This results in a limitation in the accuracy of the field profile restitution. Thus, it can deal with the guided waves in layered sandwich only when the material properties of adjacent layers do not change significantly. Despite the great efforts regarding this issue in the literature, there remain open questions. One of them is: “what is the exact threshold of contrasting material properties of adjacent layers for which this polynomial method cannot correctly restitute the roots of guided waves?” We investigated this numerical issue using the calculated guided phase velocities in 0°/φ/0°-carbon fibre reinforced plastics (CFRP) sandwich plates with gradually increasing angle φ. Then, we approached this numerical problem by varying the middle layer thickness h_90° for the 0°/90°/0°-CFRP sandwich structure, and we proposed an exact thickness threshold of the middle layer for the Legendre polynomial method limitations. We showed that the polynomial method fails to calculate the quasi-symmetric Lamb mode in 0°/φ/0°-CFRP when φ > 25°. Moreover, we introduced a new Lamb mode so-called minimum-group-velocity that has never been addressed in literature.