We use direct N-body simulations of gas embedded star clusters to study the importance of stellar collisions for the formation and mass accretion history of high-mass stars. Our clusters start in virial equilibrium as a mix of gas and proto-stars. Proto-stars then accrete matter using different mass accretion rates and the amount of gas is reduced in the same way as the mass of stars increases. During the simulations we check for stellar collisions and we investigate the role of these collisions for the build-up of high-mass stars and the formation of runaway stars. We find that a significant number of collisions only occur in clusters with initial half-mass radii rh < 0.1 pc. After emerging from their parental gas clouds, such clusters end up too compact compared to observed young, massive open clusters. In addition, collisions lead mainly to the formation of a single runaway star instead of the formation of many high mass stars with a broad mass spectrum. We therefore conclude that massive stars form mainly by gas accretion, with stellar collisions only playing a minor role if any at all. Collisions of stars in the pre-main sequence phase might however contribute to the formation of the most massive stars in the densest star clusters and possibly to the formation of intermediate-mass black holes with masses up to a few 100 Msun.