Mechanical engineers working in computer-aided design (CAD) often design components of an assembly (i.e., a collection of components with jointed connections) in isolation; loads on the isolated component are predetermined from full-assembly simulations and used to perform simulations on the component even as the component varies. However, the overall structural response of the assembly and the component loads change as the component is modified, requiring frequent full-assembly simulations using a high-fidelity model (HFM), which can be prohibitively time-consuming. We introduce the idea of a rest-of-the-assembly (RoA) reduced order model (ROM) which combines with an HFM of the designed component in order to efficiently provide full-assembly simulation feedback. The RoA (i.e., the assembly with the isolated component removed) remains unchanged throughout the component design and is thus well-approximated by a low-dimensional ROM. This decomposition of the assembly into the RoA and the designed component is effectively an instance of reduced basis element methods or domain decomposition, two methods that have been demonstrated in the field of ROM [1,2]. Training of an RoA ROM involves sampling the variation in the boundary conditions (BCs) that are applied at the interface between the RoA and the component to approximate the full-assembly behaviour for a changing component. In contrast to component-based ROM methods [3], wherein BCs are sampled exhaustively, the CAD context requires the RoA ROM to be constructed adaptively on-the-fly as the designer works hence we must limit the number of training solves (as they use the full-assembly HFM). Our novel RoA ROM training scheme includes a virtual rigid body joint formulation to enable jointed assembly simulation, inertial relief to handle potential ill-posedness of the RoA due to removal of the component, and an efficient BC sampling procedure. We demonstrate the methods on realistic mechanical assemblies.