The flexible boundary condition method (FBCM) is a well-established method for the efficient study of complex non-linear atomistic defects while avoiding finite-size effects. The method uses lattice Green's functions (LGFs) to effectively embed an atomistic domain in an infinite elastic material. Here, the FBCM is analyzed carefully and it is shown that, even in elastic linear systems where the true solution is unique, the solution of a posed problem depends on (i) the initial configuration at the start of the iterative solution process and (ii) the transition from the LGF to the continuum Green's function that is necessary because computation and storage of the infinite-domain LGF is not possible. The largest errors arise outside the atomistic domain, having implications for the application of the FBCM within multiscale models. These results show limitations in the accuracy of the FBCM that have not been previously recognized.