  The upper critical field of NaFe$_{1-x}$Co$_x$As was measured from $x=0$ to
$x=0.08$, with the magnetic field applied parallel ($H_{\rm c2}^\parallel$) and
normal ($H_{\rm c2}^\perp$) to the planes. The data were fitted using one-band
and two-band models. The orbital and paramagnetic components of the upper
critical field were extracted for $H_{\rm c2}^\parallel$, which we find to to
be strongly dominated by paramagnetic pair-breaking. In the parent compound the
paramagnetic limit is equal to the value expected by BCS theory. However,
substitution of Fe by Co leads to an enhancement above the BCS limit by a
factor of approximately 1.6. In the over-doped region, we observe a significant
convex curvature in $H_{\rm c2}^\perp$ at low temperatures, which we attribute
to the two-band nature of the superconductivity and the asymmetry between the
two bands. Additionally, we discuss the behavior of critical field anisotropy,
coherence length $\xi$, and the penetration depth $\lambda$.
