  While the spin liquid state in the frustrated pyrochlore
Tb$_{2+x}$Ti$_{2-x}$O$_{7+y}$ has been studied both experimentally and
theoretically for more than two decades, no definite description of this
unconventional state has been achieved. Using synchrotron based THz
spectroscopy in combination with quantum numerical simulations, we highlight a
significant link between two previously unrelated features: the existence of a
quadrupolar order following an ice rule and the presence of strong
magneto-elastic coupling in the form of hybridized Tb$^{3+}$ crystal-field and
phonon modes. The magnitude of this so-called vibronic process, which involves
quadrupolar degrees of freedom, is significantly dependent on small
off-stoichiometry $x$ and favors all-in all-out like correlations between
quadrupoles. This mechanism competes with the long range ordered quadrupolar
ice, and for slightly different stoichiometry, is able to destabilize it.
