One major issue that hinders the use of performance analysis in industrial design processes is the pessimism inherent to any analysis technique that applies to realistic system models. Indeed, such analyses may conservatively declare unschedulable systems that will in fact never miss any deadlines. We advocate the need to compute not only tight upper bounds on worst-case behaviors but also tight lower bounds. As a first step, we focus on uniprocessor systems executing a set of sporadic or periodic hard real-time task chains. Each task has its own priority, and the chains are scheduled according to the fixed-priority pre-emptive scheduling policy. Computing the worst-case end-to-end latency (WCEL) of each chain is complex because of the intricate relationship between the task priorities. Compared to the state of the art, our analysis provides upper bounds on the WCEL in the more general case of asynchronous task chains, and also provides lower bounds on the WCEL both for synchronous and asynchronous chains. Our computed lower bounds correspond to actual system executions exhibiting a behavior that is as close to the worst case as possible, while all other approaches rely on simulations. Extensive experiments show the relevance of lower bounds on the worst-case behavior for the industrial design of real-time embedded systems.