We address the problem of computing bounds on the worst-case end-to-end delay for a flow traversing a tandem of FIFO-multiplexing network nodes. Numerical solution methods are required, as closed-form delay bound expressions are unknown except for few specific cases. For the most accurate method, called the Least Upper Delay Bound, we present novel exact and approximate numerical algorithms, and discuss their accuracy and computation cost. Our results show that accurate computation is affordable for tandems of up to few ten nodes, and that our bounds are exponentially better than those given by the other comparable method known so far, i.e. per-node analysis.