The crystallographic and magnetic properties of the ${\mathrm{Y}}_{3}{\mathrm{Ni}}_{13\ensuremath{-}x}{\mathrm{Co}}_{x}{\mathrm{B}}_{2}$ series of compounds, with the ${\mathrm{Nd}}_{3}{\mathrm{Ni}}_{13}{\mathrm{B}}_{2}$-type structure and $x\ensuremath{\leqslant}5$, were investigated by powder x-ray diffraction and magnetic measurements. The susceptibility and magnetization data show that ${\mathrm{Y}}_{3}{\mathrm{Ni}}_{13}{\mathrm{B}}_{2}$ is a weak itinerant antiferromagnet with a N\'eel temperature ${T}_{N}=68\phantom{\rule{0.3em}{0ex}}\mathrm{K}$. A weak ferromagnetic component appears at ${T}_{N}$, suggesting uncompensated antiferromagnetic structure. The compounds with $x=0.5$ and 1, also order antiferromagnetically, with ${T}_{N}=94$ and $106\phantom{\rule{0.3em}{0ex}}\mathrm{K}$, respectively. The intrinsic weak ferromagnetic component increases with the cobalt content and a field-induced transition, from the antiferromagnetic state to a ferromagnetic state, is observed at the critical transition fields 5.6 and $1.6\phantom{\rule{0.3em}{0ex}}\mathrm{T}$, at $5\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ for $x=0.5$ and 1, respectively. A heterogeneous ferromagnetic state and in-plane magnetic anisotropy are determined for the compounds in the $2\ensuremath{\leqslant}x\ensuremath{\leqslant}5$ composition range. The Curie temperature and spontaneous magnetization increase linearly with the cobalt content. Based on the experimental data, the $T\text{\ensuremath{-}}x$ phase diagram is constructed for $x\ensuremath{\leqslant}5$. The effect of spin fluctuations is highlighted in the nickel-rich compounds in the 3:13:2 and related systems.