Moving away from fossil fuels towards renewable sources requires system operators to determine the capacity of distribution systems to safely accommodate green and distributed generation (DG). However, the DG capacity of a distribution system is often underestimated due to either overly conservative electrical demand and DG output uncertainty modelling or neglecting the recourse capability of the available components. To improve the accuracy of DG capacity assessment, this paper proposes a distributionally adjustable robust chance-constrained approach that utilises uncertainty information to reduce the conservativeness of conventional robust approaches. The proposed approach also enables fast-acting devices such as inverters to adjust to the real-time realisation of uncertainty using the adjustable robust counterpart methodology. To achieve a tractable formulation, we first define uncertain chance constraints through distributionally robust conditional value-at-risk (CVaR), which is then reformulated into convex quadratic constraints. We subsequently solve the resulting large-scale, yet convex, model in a distributed fashion using the alternating direction method of multipliers (ADMM). Through numerical simulations, we demonstrate that the proposed approach outperforms the adjustable robust and conventional distributionally robust approaches by up to 15% and 40%, respectively, in terms of total installed DG capacity.