The forage intake rate of grazing ungulates is limited either by the rate at which they encounter food items, or the rate at which food items are handled. Whether an ungulate is encounter- or handling-limited influences spatial and temporal depletion of forage, daily time budgets, and ultimately animal condition. Previously, vegetation abundance has been used as a surrogate for an ungulate's encounter rate with food items and related to observed bite rate to determine whether intake rate is encounter- or handling-limited. In temperate climates snow accumulation during winter limits access to vegetation by forcing animals to wade and paw through snow to consume underlying vegetation, increasing the amount of time required to encounter a food item. As a result, an ungulate may be handling-limited when foraging in a high biomass system under snow-free conditions, but becomes encounter-limited when snow accumulates. We derived a model that provides a frame work for estimating the rate at which a grazing ungulate encounters vegetation by considering foraging velocity, vegetation biomass and the time required to paw away snow when present. We then used data from focal observations of 36 wild elk Cervus canadensis wintering on a montane grassland in the Canadian Rockies of Alberta, Canada, to apply our model and estimate encounter rate over a range of vegetation abundance and snow conditions. Using AICc in a model selection approach we found that an asymptotic regression model of observed bite rate as a function of estimated encounter rate provided a better fit than similar models using only vegetation abundance as the explanatory variable. An asymptotic model suggests elk were handling-limited in the absence of snow, but became encounter-limited when snow accumulated. Our results demonstrate the importance of considering the influence of factors other than vegetation abundance on the intake rate of grazing ungulates.