The competition between magnetism and superconductivity was investigated for the recently discovered R${\mathrm{Ni}}_{2}$${\mathrm{B}}_{2}$C superconductors with the magnetic rare-earth elements R=Tm,Er,Ho,Dy. The systematic decrease of ${\mathit{T}}_{\mathit{c}}$, approximately scaled by the de Gennes factor, implies a very weak coupling between the rare-earth magnetic moments and the conduction electrons due to a small conduction-electron density at the rare-earth site. Associated with the antiferromagnetic order of the rare-earth moments, a pronounced dip structure in the upper critical field ${\mathit{H}}_{\mathit{c}2}$(T) was observed; a similar structure has been seen in the previously known magnetic superconductors R${\mathrm{Rh}}_{4}$${\mathrm{B}}_{4}$ and R${\mathrm{Mo}}_{6}$${\mathrm{S}}_{8}$. For ${\mathrm{HoNi}}_{2}$${\mathrm{B}}_{2}$C, the pair-breaking associated with the magnetic transition is strong enough to bring about a resistive reentrant behavior even under zero field, an effect which, to our knowledge, has not previously been observed in antiferromagnetic superconductors.