Hyperbolicity measures, in terms of (distance) metrics, how close a given graph is to being a tree. Due to its relevance in modeling real-world networks, hyperbolicity has seen intensive research over the last years. Unfortunately, the best known algorithms for computing the hyperbolicity number of a graph (the smaller, the more tree-like) have running time $O(n^4)$, where $n$ is the number of graph vertices. Exploiting the framework of parameterized complexity analysis, we explore possibilities for "linear-time FPT" algorithms to compute hyperbolicity. For instance, we show that hyperbolicity can be computed in time $O(2^{O(k)} + n +m)$ ($m$ being the number of graph edges) while at the same time, unless the SETH fails, there is no $2^{o(k)}n^2$-time algorithm.