Coherent enhancement is a powerful mechanism for improving the sensitivity of a wide range of detectors, but its practical use is often limited by the difficulty of preparing the required quantum states. We show that this difficulty has a fundamental origin: coherent enhancement of a signal interacting with a detector is quantitatively constrained by entanglement. We prove general bounds on how the strength of coherent effects can scale with system size, as a function of the single-subsystem entanglement entropy of the detector. These bounds smoothly connect the incoherent and fully coherent regimes, and apply both to parameter-estimation problems and to scattering processes. We discuss these results from two complementary perspectives: First, they appear as bounds on the quantum Fisher information of many-body states, which translate directly into limits on parameter sensitivity via the quantum Cram{'e}r-Rao bound. Second, for scattering experiments, we separate coherent enhancement of generic scattering probabilities from the distinguishability of post-scattering detector states. We bound the latter in terms of entanglement, leading to limits on how minimum detectable interaction strengths can scale with target size. Together, these results provide a unified view of coherent enhancement in metrology and scattering experiments, and motivate the development of new techniques for generating entangled detector states.