  We calculate the cold neutron-deuteron ($nd$) capture cross section,
$\sigma_{nd}$, to next-to-next-to leading order (NNLO) using the
model-independent approach of pionless effective field theory
(EFT($\pi\!\!\!/$)). At leading order we find $\sigma_{nd} = 0.315 \pm 0.217$
mb, while the experimental result is 0.508(15) mb [Jurney, Bendt and Browne in
Phys. Rev. C 25, 2810 (1982)] for a laboratory neutron velocity of 2200 m/s. At
next-to-leading-order (NLO), we show that $\sigma_{nd}$ is sensitive to the low
energy constant (LEC), $L_1^{(0)}$, of the two-nucleon isovector current
appearing at NLO. A fit of $L_1^{(0)}$ at NLO to the triton magnetic moment
yields a NLO prediction of $\sigma_{nd}=0.393 \pm 0.164$ mb, where the error
comes from propagating the error from the $L_1^{(0)}$ fit. At
next-to-next-to-leading-order (NNLO), we find that a new three-nucleon magnetic
moment counterterm is required for renormalization group invariance of both
$\sigma_{nd}$ and the triton magnetic moment. Fitting the NNLO correction to
$L_1^{(0)}$ (denoted $L_1^{(1)}$) to cold neutron-proton capture
($\sigma_{np}$) yields a NNLO prediction of $\sigma_{nd}=0.447 \pm 0.130$ mb,
where the error comes from propagating the error from the $L_1^{(1)}$ fit. We
also study different fittings of $L_1^{(0)}$ and $L_1^{(1)}$ to $\sigma_{np}$,
$\sigma_{nd}$, and/or the triton magnetic moment. For example, fitting
$L_1^{(0)}$ simultaneously to $\sigma_{np}$, $\sigma_{nd}$, and the triton
magnetic moment at NLO, and fitting $L_1^{(1)}$ simultaneously to $\sigma_{np}$
and $\sigma_{nd}$ at NNLO, yields $\sigma_{nd} = 0.480 \pm 0.114$ mb and $0.511
\pm 0.042$ mb, respectively, where errors are naively estimated from
EFT($\pi\!\!\!/$) power counting. In addition, we discuss how Wigner-SU(4)
symmetry may alter the naive EFT($\pi\!\!\!/$) expansion of $\sigma_{nd}$.
