Summary: This paper is concerned with a parabolic-parabolic Keller-Segel-type system in a bounded domain \(\Omega\subset\mathbb{R}^N\) (with \(N=2\) or \(N=3\)) presenting source and damping terms. We impose Neumann and Robin boundary conditions to each one of the two unknowns of the problem and study the non-negative solutions which blow up in finite time \(t^{\ast}\). In this way, it is possible to derive explicit lower bounds for \(t^{\ast}\), under appropriate conditions on the data of the problem.