Gravitational models of self-tuning are those in which vacuum energy has no observable effect on spacetime curvature, even though it is a priori unsuppressed below the cutoff. We complement Weinberg's no-go theorem by studying field-theoretic completions of self-adjustment allowing for broken translations as well as other generalizations, and identify new obstructions. Our analysis uses a very general Källén-Lehmann spectral representation of the exchange amplitude for conserved sources of energy-momentum and exploits unitarity and Lorentz invariance to show that a transition from self-tuning of long wavelength sources to near general relativity (GR) on shorter scales is generically not possible. We search for novel ways around our obstructions and highlight two interesting possibilities. The first is an example of a unitary field configuration on anti-de Sitter space with the desired transition from self-tuning to GR. A second example is motivated by vacuum energy sequestering.