How nonlinear systems dynamically respond to external perturbations essentially determines their function. Weak perturbations induce response dynamics near a stable operating point, often approximately characterized by linear response theory. However, stronger driving signals may induce genuinely nonlinear responses, including tipping transitions to qualitatively different dynamical states. Here, we analyze how inter-unit coupling impacts responses to periodic perturbations. We find that already in minimal systems of two identical and linearly coupled units, coupling impacts the dynamical responses in a distinct way. Any non-zero coupling strength extends the regime of non-tipping local responses relative to uncoupled units. Intriguingly, finite coupling may be more effective than infinitely strong coupling in keeping responses from tipping. Interestingly, already weak coupling may create novel response modes in strongly driven systems, implying the existence of multiple tipping points instead of only one. These results persist for systems of non-identical units, systems with nonlinear coupling, and larger networks of coupled units.