This paper deals with a system of two coupled partial differential equations arising in chemotaxis, involving nonlinear diffusion and nonlinear and signal-dependent sensitivity. Depending on the interplay between such nonlinearities, we establish the existence of global classical solutions which are uniformly bounded in time. Precisely, we study the zero-flux chemotaxis-system {u(t) = del center dot ((u + 1)(m-1) del u - u(u + 1)(alpha-1)chi(v)del v) in Omega x (0,infinity), 0 = Delta v - v + u in Omega x (0,infinity), (lozenge) omega being a bounded and smooth domain of R-n, n >= 1, and where m, alpha is an element of R, with alpha <= max{m, m+1/2}. Additionally, 0 < chi is an element of C-1((0, infinity)) obeys the inequality chi(s) <= chi 0/s(k), for some chi(0) > 0, k >= 1 and all s > 0. We prove that for any nonnegative and properly regular initial data u(x, 0), the initial-boundary value problem associated to (lozenge) admits a unique globally bounded classical solution, provided some smallness assumptions on chi(0) are satisfied. In addition, in this article we compare our results with those achieved in the recent paper (Wang et al. in J Differ Equ 263(5):2851-2873, 2017); we will emphasize how the employment of independent techniques used to solve problem (lozenge) may lead to complementary conclusions.