We present the results of a systematic first-principles investigation of the requirements for developing realistic and reliable structural models for amorphous tetrahedral carbon (a-tC) and relate those structural models to the physical properties of this material. Within a linear combination of atomic orbitals formulation of density functional theory, we show that a large variational flexibility is required to accurately treat the highly defected and strained structures that can exist in a-tC. The average strain in the a-tC lattice is predicted to be roughly twice the strain of having all carbon atoms in three-member rings. A key figure of merit of a structural model, the proportion of three-fold bonded atoms, is shown to triple in going from a minimal basis description of a structure to a high quality basis. The basis-converged calculations agree well with experimental observables, such as the presence of four-member rings, lack of dangling bonds, and a significant gap. The simulations predict a much larger proportion of three-fold atoms than estimated in simple analyses of EELS and neutron scattering experiments. We show that the larger three-fold fraction is indeed consistent with the properties of a-tC, and imply that there are flaws in the simplifying assumptions that go into constructing experimental estimates of coordination numbers. These results highlight the perils of applying highly simplified theoretical models for a-tC before the correct physics has been identified and built into the models.