Single-walled carbon nanotubes theoretically possess ultimate intrinsic tensile strengths in the 100-200 GPa range, among the highest in existing materials. However, all of the experimentally reported values are considerably lower and exhibit a considerable degree of scatter, with the lack of structural information inhibiting constraints on their associated mechanisms. Here, we report the first experimental measurements of the ultimate tensile strengths of individual structure-defined, single-walled carbon nanotubes. The strength depends on the chiral structure of the nanotube, with small-diameter, near-armchair nanotubes exhibiting the highest tensile strengths. This observed structural dependence is comprehensively understood via the intrinsic structure-dependent inter-atomic stress, with its concentration at structural defects inevitably existing in real nanotubes. These findings highlight the target nanotube structures that should be synthesized when attempting to fabricate the strongest materials.