Two grades of commercial, highly graphitic, vapor grown carbon nanofibers Pyrograf-III, that were pyrolytically stripped and heat treated, with an average outer diameter of 150 nm, were tested for their tensile strength by a MEMS-based mechanical testing platform. The applied force was measured by a surface micromachined loadcell whose deflection was extracted by digital image correlation with an accuracy of 50 nm or better. The mean strength of pyrolytically-stripped nanofibers was 2.93 GPa while the average strength of heat-treated nanofibers was 2.54 GPa. The Weibull parameters extracted from experimental data showed significant randomness in the flaw population in both types of fibers. High-resolution SEM images of matching ruptured surfaces showed purely brittle fracture, and, in some occasions, distinct partial "sword-in-sheath" type of failure. The latter was potentially due to mutual sliding of graphene planes during fiber fracture, which was attributed to the "dixie cup" structure of Pyrograf-lll. This is the first experimental work that addresses the mechanics of carbon nanofibers at the individual fiber scale. To date their strength has been assumed to be similar to conventional carbon fibers.