  The dynamics of an open system crucially depends on the correlation function
of its environment, $C_B(t)$. We show that for thermal non-Harmonic
environments $C_B(t)$ may not decay to zero but to an offset, $C_0>0$. The
presence of such offset is determined by the environment eigenstate structure,
and whether it fulfills or not the eigenstate thermalization hypothesis.
Moreover, we show that a $C_0>0$ could render the weak coupling approximation
inaccurate and prevent the open system to thermalize. Finally, for a realistic
environment of dye molecules, we show the emergence of the offset by using
matrix product states (MPS), and discuss its link to a 1/f noise spectrum that,
in contrast to previous models, extends to zero frequencies. Thus, our results
may be relevant in describing dissipation in quantum technological devices like
superconducting qubits, which are known to be affected by such noise.
