The relationship between community complexity and stability has been the subject of an enduring debate in ecology over the last 50 years. Results from early model communities showed that increased complexity is associated with decreased local stability. I demonstrate that increasing both the number of species in a community and the connectance between these species results in an increased probability of local stability in discrete-time competitive communities, when some species would show unstable dynamics in the absence of competition. This is shown analytically for a simple case and across a wider range of community sizes using simulations, where individual species have dynamics that can range from stable point equilibria to periodic or more complex. Increasing the number of competitive links in the community reduces per-capita growth rates through an increase in competitive feedback, stabilising oscillating dynamics. This result was robust to the introduction of a trade-off between competitive ability and intrinsic growth rate and changes in species interaction strengths. This throws new light on the discrepancy between the theoretical view that increased complexity reduces stability and the empirical view that more complex systems are more likely to be stable, giving one explanation for the relative lack of complex dynamics found in natural systems. I examine how these results relate to diversity-biomass stability relationships and show that an analytical solution derived in the region of stable equilibrium dynamics captures many features of the change in biomass fluctuations with community size in communities including species with oscillating dynamics.