The just-noticeable difference (DeltaI) in the intensity (I) of sound is typically reported to be a constant or a slightly decreasing ratio of the baseline intensity (known as Weber's law, and the "near-miss to Weber's law", respectively). However, in the relatively few studies on the intensity discrimination of very brief sounds, DeltaI/I is usually found to be non-monotonic, with poorest discrimination in the middle of the auditory dynamic range. Here, it is demonstrated that this "severe departure from Weber's law" or "mid-level hump" is not merely a phenomenon of short-duration sounds. In normal-hearing subjects (n=8), the near-miss to Weber's law that is observed with the discrimination of 300 ms-long, 4 kHz tones, gives way to a significant mid-level hump if tone intensities are not fixed over a great many trials (as is standard practice) but are instead randomly roved, trial-to-trial, over a wide intensity range. This was not the case in subjects with mild to moderate hearing impairment (n=4). Furthermore, in the discrimination of widely-roved, 4 ms-long, 4 kHz tone bursts, the performance of normal-hearing subjects did not significantly worsen at mid-levels compared to the unroved condition, unlike what was found with the 300 ms-long tones. It is suggested that mid-level humps could simply be the product of the well-known mid-level compressive nonlinearity in cochlear mechanics. We further suggest that the hump is eliminated, and the near-miss to Weber's law is produced, by a more central mechanism such as the recently reported "adaptation to sound-level statistics", which is bypassed during wide-range roving and possibly when sound durations are brief.