The viability of long-distance telesurgery hinges on reliable network Quality of Service (QoS), yet the impact of realistic network degradations on task performance is not sufficiently understood. This paper presents a comprehensive analysis of how packet loss, delay, and communication loss affect telesurgical task execution. We introduce NetFI, a novel fault injection tool that emulates different network conditions using stochastic QoS models informed by real-world network data. By integrating NetFI with a surgical simulation platform, we conduct a user study involving 15 participants at three proficiency levels, performing a standardized Peg Transfer task under varying levels of packet loss, delay, and communication loss. We analyze the effect of network QoS on overall task performance and the fine-grained motion primitives (MPs) using objective performance and safety metrics and subjective operator's perception of workload. We identify specific MPs vulnerable to network degradation and find strong correlations between proficiency, objective performance, and subjective workload. These findings offer quantitative insights into the operational boundaries of telesurgery. Our open-source tools and annotated dataset provide a foundation for developing robust and network-aware control and mitigation strategies.