  We study the cosmological consequences of a class of Dirac-Born-Infeld
models, and assess their viability as a candidate for the recent acceleration
of the Universe. The model includes both the rolling tachyon field and the
generalized Chaplygin gas models as particular limits, and phenomenologically
each of these provides a possible mechanism for a deviation of the value of the
dark energy equation of state from its canonical (cosmological constant) value.
The field-dependent potential that is characteristic of the rolling tachyon
also leads to variations of the fine-structure constant $\alpha$, implying that
the model can be constrained both by standard cosmological probes and by
astrophysical measurements of $\alpha$. Our analysis, using the latest
available low-redshfit data and local constraints from atomic clock and weak
equivalence principle experiments, shows that the two possible deviations of
the dark energy equation of state are constrained to be
$\log_{10}{(1+w_0)_V}<-7.85$ and $\log_{10}{(1+w_0)_C}<-0.85$, respectively for
the rolling tachyon and Chaplygin components, both being at the $95.4\%$
confidence level (although the latter depends on the choice of priors, in a way
that we quantify). Alternatively, the $95.4\%$ confidence level bound on the
dimensionless slope of the potential is $\log_{10}{\lambda}<-5.36$. This
confirms previous analyses indicating that in these models the potential needs
to be extremely flat.
