The speech intelligibility index (SII) calculation is based on the assumption that the effective range of signal-to-noise ratio (SNR) regarding speech intelligibility is [-15 dB; +15 dB] with an importance equally distributed over this interval. In a specific frequency band, speech intelligibility would no longer be influenced if the SNR is increased above +15 dB or decreased below -15 dB. These assumptions were tested in four experiments measuring speech reception thresholds (SRTs) with a speech target and speech-spectrum noise, while attenuating target or noise above or below 1400 Hz, with different levels of attenuation allowing the testing of different SNRs in the two bands. The results confirmed that SRT varied linearly with attenuation at low-attenuation levels and an asymptote was reached for high-attenuation levels. However, this asymptote was reached (intelligibility was not influenced by further attenuation) for different attenuation levels across conditions. The -15-dB SII limit was confirmed for high-pass filtered targets, whereas for low-pass filtered targets, intelligibility was further impaired by decreasing the SNR below -15 dB (until about -30 dB). For the high-pass and low-pass filtered noises, speech intelligibility kept improving when increasing the SNR beyond the +15 dB (up to about 40 dB). Before reaching the asymptote, a 10-dB increase of SNR obtained by filtering the noise had a larger impact on intelligibility than a corresponding 10-dB decrease of SNR obtained by filtering the target. These results question the use of the SNR range and the importance function adopted by the SII when considering sharply filtered signals.