About a century ago, Jeans (1919) discovered that if binary stars reach a state approximating energy equipartition, for example, through many dynamical encounters that exchange energy, their eccentricity distribution can be described by dN/de = 2e. This is referred to as the thermal eccentricity distribution, and has been widely used for initial conditions in theoretical investigations of binary stars. However, observations suggest that the eccentricity distributions of most observed binaries, and particularly those with masses less than or similar to 5 M-circle dot, are flatter than thermal and follow more closely to a uniform distribution. Nonetheless, it is often argued that dynamical interactions in a star cluster would quickly thermalize the binaries, which could justify imposing a thermal eccentricity distribution at birth for all binaries. In this paper, we investigate the validity of this assumption. We develop our own rapid semi analytic model for binary evolution in star clusters, and also compare it with detailed N-body and Monte Carlo star cluster models. We show that, for nearly all binaries, dynamical encounters fail to convert an initially uniform eccentricity distribution to thermal within a star cluster's lifetime. Thus, if a thermal eccentricity distribution is observed, it is likely imprinted upon formation rather than through subsequent long-term dynamical processing. Theoretical investigations that initialize all binaries with a thermal distribution will make incorrect predictions for the evolution of the binary population. Such models may overpredict the merger rate for binaries with modest orbital separations by a factor of about two.