Snow is a sensitive component of the climate system. In many parts of the world, water stored as snow is a vital resource for agriculture, tourism and the energy sector. As uncertainties in climate change assessments are still relatively large, it is important to investigate the interdependencies between internal climate variability and anthropogenic climate change and their impacts on snow cover. We use regional climate model data from a new single-model large ensemble with 50 members (ClimEX LE) as a driver for the physically based snow model SNOWPACK at eight locations across the Swiss Alps. We estimate the contribution of internal climate variability to uncertainties in future snow trends by applying a Mann-Kendall test for consecutive future periods of different lengths (between 30 and 100 years) until the end of the 21st century. Under RCP8.5, we find probabilities between 10% and 60% that there will be no significant negative trend in future mean snow depths over a period of 50 years. While it is important to understand the contribution of internal climate variability to uncertainties in future snow trends, it is likely that the variability of snow depth itself changes with anthropogenic forcing. We find that relative to the mean, interannual variability of snow increases in the future. A decrease in future mean snow depths, superimposed by increases in interannual variability, will exacerbate the already existing uncertainties that snow-dependent economies will have to face in the future.