We describe the design, construction and performance of three realizations of a high-field superconducting magnetic trap for neutral atoms and molecules. Each of these traps utilizes a pair of coaxial coils in the anti-Helmholtz geometry and achieves depths greater than 4 T, allowing it to capture magnetic atoms and molecules cooled in a cryogenic buffer gas. Achieving this depth requires that the repulsive force between the coils (which can exceed 30 metric tons) be contained. We also describe additional features of the traps, including the elimination of trapped fluxes from the coils and the integration of the coils into a cryogenic vacuum environment suitable for producing cold atoms and molecules.