  Cosmological $\alpha$-attractors are a compelling class of inflationary
models. They lead to universal predictions for large-scale observables, broadly
independent from the functional form of the inflaton potential. In this work we
derive improved analytical predictions for the large-scale observables, whose
dependence on the duration of reheating and the parameter $\alpha$ is made
explicit. We compare these with Planck and BICEP/Keck 2018 data in the
framework of a Bayesian study, employing uniform logarithmic and linear priors
for $\alpha$. Our improved universal predictions allow direct constraints on
the duration of reheating. Furthermore, while it is well-known that CMB
constraints on the tensor-to-scalar ratio can be used to place an upper bound
on the $\alpha$ parameter, we demonstrate that including the
$\alpha$-dependence of the scalar spectral tilt yields novel constraints on
$\alpha$. In particular, for small $\alpha$, the scalar spectral tilt scales
with $\log_{10}\alpha$, regardless of the specific potential shape. For
decreasing $\alpha$, this eventually puts the models in tension with CMB
measurements, bounding the magnitude of $\alpha$ from below. Therefore, in
addition to the upper bound from the tensor-to-scalar ratio, we derive the
first lower bound on the magnitude of $\alpha$ for $\alpha$-attractor T-models,
$\log_{10}{\alpha} = -4.2^{+5.4}_{-8.6}$ at $95\%$ C.L. .
