How do deleterious mutations interact to affect fitness? The answer to this question has substantial implications for a variety of important problems in population biology, including the evolution of sex, the rate of adaptation and the conservation of small populations. Here we analyse a mathematical model of competition for food in which deleterious mutations affect competitive ability. We show that, if individuals usually compete in small groups, then competition can easily lead to a type of genetic interaction known as synergistic epistasis. This means that a deleterious mutation is most damaging in a genome that already has many other deleterious mutations. We also show that competition in small groups can produce a large advantage for sexual populations, both in mean fitness and in ability to resist invasion by asexual lineages. One implication of our findings is that experimental efforts to demonstrate synergistic epistasis may not succeed unless the experiments are redesigned to make them much more naturalistic.