Gaussian analysis of interstellar neutral hydrogen emission profiles has revealed a pervasive broad component with a width on the order of 34kms ~(-1). When present, this component can most readily be identified in high galactic latitude directions where the H I profiles are either intrinsically weak or simple. Examination of published data reveals that this characteristic line width has been found in a variety of other H I features including compact high-velocity clouds, very-high-velocity clouds, and the Magellanic Stream. When its presence is accounted for in the analysis of H I profiles, other families of line widths at 14 and 6kms~(-1) are clearly revealed. Possible mechanisms for producing this broad background component are discussed, including temperature, turbulence, and the critical ionization velocity effect. A line width on the order of 34kms~(-1) would imply a kinetic temperature of 24,000 K, too high to keep the gas neutral; hence it should not be observed in H I emission spectra. Turbulent motions could explain a pervasive broad component, but not why it always has the same numerical value in various classes of H I emission line features. The critical ionization velocity effect hypothesis is intriguing because 34kms~(-1) is the value for helium. Clearly, this could be a coincidence but the other prominent distribution peaks correspond to two families of critical ionization velocities of abundant interstellar elements including C, N, and O (about 14kms ~(-1)) and metals (about 6kms~(-1)). Unfortunately, the mechanism by which this effect operates, even in laboratory situations, is not clearly understood. It is suggested that further investigation of the distribution of H I component line widths by allowing for the existence of a pervasive broad underlying component may cast a clearer light on this intriguing phenomenon.
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