  We introduce a class of 3d theories consisting of strongly-coupled ${\mathcal
N}=4$ systems coupled to ${\mathcal N}=3$ Chern-Simons gauge multiplets, which
exhibit ${\mathcal N}=4$ enhancements when a peculiar condition on the
Chern-Simons levels is met. An example is the $SU(N)^3$ Chern-Simons theory
coupled to the 3d $T_N$ theory, which enhances to ${\mathcal N}=4$ when
$1/k_1+1/k_2+1/k_3=0$. We also show that some but not all of these ${\mathcal
N}=4$ enhancements can be understood by considering M5-branes on a special
class of Seifert manifolds. Our construction provides a large class of
${\mathcal N}=4$ theories which have not been studied previously.
