While skilled immigration ceteris paribus provides an immediate boost to GDP per capita by adding to the human capital stock of the receiving economy, might it also reduce the number of 'good jobs', i.e. those with training, available to indigenous workers? We analyse this issue theoretically and empirically. The theoretical model shows how skilled immigration can affect the sectoral allocation of labor and how it may have a positive or negative effect on the training and social mobility of native born workers. The empirical analysis uses UK data from 2001 to 2018 to show that training rates of UK born workers have declined in a period where immigration has been rising strongly, and have declined significantly more in high wage non-traded sectors. At a more disaggregated level however this link is much less strong, though there is evidence of different training effects of skilled immigration across traded and non-traded sectors. Hiring of UK born workers in high wage non-traded sectors has also been negatively affected by skilled immigration, although this effect is not large. Taken together the theoretical and empirical analyses suggest that skilled immigration may have some role in allocating native born workers away from 'good jobs' sectors but it is unlikely to be a major driver of social mobility.