  Rolling tachyon field models are among the candidates suggested as
explanations for the recent acceleration of the Universe. In these models the
field is expected to interact with gauge fields and lead to variations of the
fine-structure constant $\alpha$. Here we take advantage of recent
observational progress and use a combination of background cosmological
observations of Type Ia supernovas and astrophysical and local measurements of
$\alpha$ to improve constraints on this class of models. We show that the
constraints on $\alpha$ imply that the field dynamics must be extremely slow,
leading to a constraint of the present-day dark energy equation of state
$(1+w_0)<2.4\times10^{-7}$ at the $99.7\%$ confidence level. Therefore current
and forthcoming standard background cosmology observational probes can't
distinguish this class of models from a cosmological constant, while detections
of $\alpha$ variations could possibly do so since they would have a
characteristic redshift dependence.
