Mechanical response of ceramic matrix composites is critically dependent on properties of fibers and fiber coatings. In the present study, we examine effects of these and other constituent properties on the tensile response of unidirectionally-reinforced composites through a combination of existing models of fiber fracture, Monte Carlo simulations of concurrent matrix and fiber fragmentation, and complementary experimental measurements. The results show that, contrary to prevailing understanding, the process of fiber fragmentation cannot go to completion in a uniaxial tensile test. At low stresses (before the maximum), fiber breaks are essentially random throughout the bundle; in contrast, at the stress maximum, all additional breaks are localized to regions close to the fracture plane (within a distance dictated by the characteristic transfer length). Therefore, the resulting distributions in fiber pullout length and strength of pulled-out fibers do not match predictions from existing models. Although the composite stress-strain response associated with fragmentation can be accurately predicted by existing fragmentation models, accurate prediction of the point of instability requires consideration of the local response of fibers in the most heavily strained regions (within matrix crack planes). The response in the latter regions leads to instability at stresses and strains that are lower than those obtained from the average stress-strain response. Additionally, the composite strength and failure strain are found to be sensitive to the fiber volume fraction and the matrix strength distribution. One implication is that the full potential of the fibers may not be realizable in composites of practical interest.