The asymmetries observed in the line profiles of solar flares can provide important diagnostics of the properties and dynamics of the flaring atmosphere. In this paper the evolution of the Hα and Ca ii λ8542 lines are studied using high spatial, temporal, and spectral resolution ground-based observations of an M1.1 flare obtained with the Swedish 1 m Solar Telescope. The temporal evolution of the Hα line profiles from the flare kernel shows excess emission in the red wing (red asymmetry) before flare maximum and excess in the blue wing (blue asymmetry) after maximum. However, the Ca ii λ8542 line does not follow the same pattern, showing only a weak red asymmetry during the flare. RADYN simulations are used to synthesize spectral line profiles for the flaring atmosphere, and good agreement is found with the observations. We show that the red asymmetry observed in Hα is not necessarily associated with plasma downflows, and the blue asymmetry may not be related to plasma upflows. Indeed, we conclude that the steep velocity gradients in the flaring chromosphere modify the wavelength of the central reversal in the Hα line profile. The shift in the wavelength of maximum opacity to shorter and longer wavelengths generates the red and blue asymmetries, respectively.