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The channel-structure-based estimation has proved its capacity to enhance the performance over channels that exhibit a sparse impulse response. In this paper, it is shown that, below a signal-to-noise ratio (SNR) limit, the channel estimation performance, based on its true impulse response structure, is no more optimal in the mean square error (MSE) sense. The orthogonal frequency-division multiplexing (OFDM) transmission framework is considered, where the MSE analysis of the channel frequency response is first elaborated. Then, the SNR threshold lower bounding the interval over which the structured channel estimation is optimal is derived. Finally, an approximation leads to a new form of the threshold, which is more suitable for prediction in practice. The analysis handles both of the block- and the comb-type channel estimation and accounts for possible spectral-shaping incorporation.