Tunnel field effect transistors (TFETs) have generated much interest in the pursuit of energy-efficient electronics due to their potential to surpass the classical subthreshold-slope limit of 60 mV/decade, allowing them to switch at much lower voltages. Heterojunctions with a broken gap (type-III) or a small staggered gap (type-II) have become promising candidates for TFETs because they are predicted to provide steep subthreshold slopes limited only by the steepness of the band-edges of the material, while providing enough current for high speed operation. However, experimental results to date have not demonstrated subthreshold swings that are steeper than the classical limit. To solve this, it is necessary to understand how sharp the band-edges can be in such materials in the presence of realistic imperfections. We have studied this for some of the most commonly proposed materials systems: InAs/GaSb (type-III) and InGaAs/GaAsSb (type-II), which we have grown via MOCVD. Two-terminal measurements are used to reveal the steepness of the band-edges and to predict an ideal subthreshold slope that is intrinsic to the material interface. We find that these interfaces are prone to the formation of misfit dislocations due to strain buildup of intermixed compositions. I-V measurements show that these dislocations, along with point defects, result in less sharp band-edges and poorer predicted subthreshold swings. We propose that point-defects gettered by dislocations lead to defect states near the band edge that lower steepness, as well as strain fields and charge that cause band alignment to change across the interface. We have identified techniques to control these defect densities, including growing the III-Sb layer at the top to prevent As-Sb swap, purposely straining III-Sb layers, and lowering growth temperature near the interface to suppress intermixing. We have also employed annealing to allow lower and/or more uniform point defect concentrations and more uniform intermixing. We show that this can obtain considerably improved steepness. Low temperature measurements reveal that steepness is not a function of temperature, even in devices that show no observable defects in cross-section TEM, indicating that in all cases, the sharpness of tunneling is limited by materials defects and inhomogeneity. Furthermore, our temperature-independence in two-terminal measurements indicate that most published TFET devices, which show a strong temperature dependence of subthreshold slope, are likely overpowered by thermally-activated parasitic effects originating from the gate oxide and channel. Our results indicate that the design of TFETs need to be modified to mitigate such parasitic effects. Furthermore, even in the absence of such parasitics, the material quality likely limits the ultimate performance of band-edge switching in TFETs, and further control of materials or a change in design is required in order to perform better than the classical limit.