Much of the evidence for the idea that individuals differ in their propensity to innovate and solve new problems has come from studies on captive primates. Increasingly, behavioural ecologists are studying innovativeness in wild populations, and uncovering links with functional behaviour and fitness-related traits. The relative importance of genetic and environmental factors in driving this variation, however, remains unknown. Here, we present the results of the first large-scale study to examine a range of causal factors underlying innovative problem-solving performance (PSP) among 831 great tits (Parus major) temporarily taken into captivity. Analyses show that PSP in this population: (i) was linked to a variety of individual factors, including age, personality and natal origin (immigrant or local-born); (ii) was influenced by natal environment, because individuals had a lower PSP when born in poor-quality habitat, or where local population density was high, leading to cohort effects. Links with many of the individual and environmental factors were present only in some years. In addition, PSP (iii) had little or no measurable heritability, as estimated by a Bayesian animal model; and (iv) was not influenced by maternal effects. Despite previous reports of links between PSP and a range of functional traits in this population, the analyses here suggest that innovativeness had weak if any evolutionary potential. Instead most individual variation was caused by phenotypic plasticity driven by links with other behavioural traits and by environmentally mediated developmental stress. Heritability estimates are population, time and context specific, however, and more studies are needed to determine the generality of these effects. Our results shed light on the causes of innovativeness within populations, and add to the debate on the relative importance of genetic and environmental factors in driving phenotypic variation within populations.