Fiber bundle models are useful tools for explaining dynamic failure behavior in heterogeneous materials. Such models shed light on diverse phenomena such as fatigue in structural materials and earthquakes in geophysical settings. Building good theoretical models has proven straightforward, but analyzing them has required delving into statistical details of the interaction of various flaw features and failure configurations, which has proven to be deceptively difficult. In this paper, we present a new method for reliability analysis of dynamic fiber bundle models. As in previous works, we assume that a fiber has a failure rate following a power law in its load level. However, unlike the exponential distribution used in the previous works, we consider that the remaining lifetimes of the surviving fibers follow Weibull distributions according to either the cumulative exposure model or tampered failure rate model. We develop both exact asymptotic reliability formulas and easy to compute bounds. Further, we also describe a fast simulation algorithm that greatly increases the bundle sizes that can be analyzed.