We present high-resolution optical spectroscopy of the narrow-line Seyfert 1 galaxy and ultrasoft X-ray source RE J1034+396. We examine the profiles of the Hα, Hβ and [Oiii] lines, and find evidence for an intermediate-velocity (FWHM~1000 km s^-1) line-emitting region which produces significant amounts of both the permitted and forbidden line fluxes. This intermediate region probably has a scale-size of order 1pc, and its emission dominates the Balmer line profiles, causing the net FWHM to be relatively low. There is also an underlying broader component to the Balmer lines which has FWHM~2500 km s^-1, and which almost certainly originates from the normal broad-line region. Since narrow Balmer lines are now a common, if not ubiquitous, feature of ultrasoft X-ray AGN, this result implies a causal link between the slope of the soft X-ray spectrum and the physical conditions in the broad- and intermediate-line region. Our observations show that AGN models with abnormally low-mass central black holes, or broad-line clouds at abnormally large radii from the black hole, are not a requirement to explain RE J1034+396.