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We establish the existence of solutions to systems of second-order dynamic equations on time scales with the right member f, a Δ-Carathéodory function. First, we consider the case where the nonlinearity f does not depend on the Δ-derivative, xΔ(t). We obtain existence results for Strum-Liouville and for periodic boundary conditions. Finally, we consider more general systems in which the nonlinearity f depends on the Δ-derivative and satisfies a linear growth condition with respect to xΔ(t). Our existence results rely on notions of solution-tube that are introduced in this paper.