A sound that is briefly interrupted by a silent gap is perceived as discontinuous. However, when the gap is filled with noise, the sound may be perceived as continuing through the noise. It has been shown that this continuity illusion depends on the masking of the omitted target sound, but the underlying mechanisms have yet to be quantified thoroughly. In this article, we systematically quantify the relation between perceived continuity and the duration, relative power, or notch width of the interrupting broadband noise for interrupted and non-interrupted amplitude-modulated tones at different frequencies. We fitted the psychometric results in order to estimate the range of the noise parameters that induced auditory grouping. To explain our results within a common theoretical framework, we applied a power spectrum model to the different masking results and estimated the critical bandwidth of the auditory filter that may be responsible for the continuity illusion. Our results set constraints on the spectral resolution of the mechanisms underlying the continuity illusion and provide a stimulus set that can be readily applied for neurophysiological studies of its neural correlates.