The problem of cascading failures in cyber-physical networks is garnering much attention for different network models underlining various applications. While a variety of analytic results has been reported for the case of large networks, very few of them are readily applicable to finite-size networks. This paper studies cascading failures in finite-size geometric networks where the number of nodes is on the order of tens or hundreds as in many real-life networks. First, the impact of the tolerance parameter on network resiliency is investigated. We quantify the network reaction to initial disturbances of different sizes by measuring the damage imposed on the network. Lower and upper bounds on the number of failures are derived to characterize such damages. In addition to the finite analysis, an asymptotic analysis of both bounds is carried out, discovering a threshold behavior of the network as the tolerance parameter changes. The critical value of the tolerance parameter in the asymptotic regime is further derived. Findings of this paper, in particular, shed light on how to choose the tolerance parameter appropriately such that a cascade of failures could be avoided.