The presence of giant shells or loops in giant H II regions is clear witness to the mechanical energy input from massive stars. Here we evaluate the impact that winds may have on the structure of giant nebulae and on their supersonic velocity dispersion. We follow the recent suggestion of Chu and Kennicutt to see whether a combination of a large number of unresolved wind-driven shells caused by massive stars could produce the integrated broad Gaussian profiles typical of giant H II regions. The results, accounting for a wide range of energies, densities, and velocities or ages of the expanding shells, show that supersonic Gaussian profiles may arise only from a collection of unresolved wind-driven shells if the shells present a peculiar velocity distribution which implies a strongly peaked age distribution leading to an awkward star formation history. On the other hand, a uniform distribution of ages produces profiles with a flat-topped core defined by the terminal shell velocity and a steep decay as ν-6, up to the largest detectable shell speed. Thus, supersonic profiles can arise only if the final speed of the unresolved shells is supersonic. This implies an equally supersonic random speed of motions in the ionized gas disrupting the shells before they slow down to subsonic speeds. It also implies a mechanism, independent of the shells caused by massive stars, responsible for the supersonic stirring of the background medium. These facts, together with the conditions for shells to remain unresolved by present-day devices (energies, final speeds, and ages), indicate that the winds may be produced by low-mass stars. In the latter case, if the sources move supersonically in the gravitational potential of the whole system, they could stir the gas, with their cometary bow shocks, to a velocity dispersion σgas σstars, causing a supersonic local random speed of motions within the system.