We study the lowest achievable mean-square estimation error in two limiting optimal linear filtering problems. First, when the intensity of the process noise tends to zero, the lowest achievable mean-square estimation error is a function of the unstable poles of the system. Second, when the intensity of the measurement noise tends to zero, the lowest achievable mean-square estimation error is a function of the nonminimum phase zeros of the system. We link these results with Bode integral characterisations of performance limitations in linear filtering.