  It is notoriously difficult to make quantitative theoretical predictions of
the superconducting $T_c$, either from first-principles or even from a
knowledge of normal state properties. Ultimately, this reflects the fact that
the energy scales involved in the superconducting state are extremely small in
natural units, and that $T_c$ depends exponentially on a subtle interplay
between different interactions so that small uncertainties in microscopic
processes can lead to order 1 effects on $T_c$. However, in some circumstances,
it may be possible to determine (approximate) bounds on $T_c$. Here, we propose
such a bound for the conventional phonon-mediated mechanism of pairing with
strongly retarded interactions, i.e. in the case in which $\hbar\bar \omega \ll
E_F$ where $\bar \omega$ is an appropriate characteristic phonon frequency and
$E_F$ is the Fermi energy. Specifically, drawing on both empirical results
(shown in Figure 2 below) and recent results[1] of determinant quantum Monte
Carlo (DQMC) studies of the paradigmatic Holstein model, we propose that
\begin{equation}
  k_B T_c \leq A_{max} \ \hbar \bar \omega \end{equation} where $A_{max}$ is a
dimensionless number of order one that we estimate to be \begin{equation}
  A_{max} \approx 1/10. \end{equation}
