Large, high-energy ($E>100\text{ }\text{ }\mathrm{GeV}$) cosmic neutrino telescopes are now quite mature. IceCube, for example, observes about 50 000 well-reconstructed single atmospheric neutrino events/year, with energies above 100 GeV. Although the neutrino detection probability is small, current detectors are large enough so that it is possible to detect two neutrinos from the same cosmic-ray interaction. In this paper, we calculate the expected rate of double-neutrino interactions from a single cosmic-ray air shower. The rate is small, about $0.07\text{ }\text{ }\mathrm{events}/\mathrm{year}$ for a $1\text{ }\text{ }{\mathrm{km}}^{3}$ detector like IceCube, with only a small dependence on the assumed cosmic-ray composition and hadronic interaction model. For a larger detector, like the proposed KM3Net, the rate is about $0.8\text{ }\text{ }\mathrm{events}/\mathrm{year}$, high enough to be easily observable. These double neutrino interactions are the major irreducible background to searches for pairs of supersymmetric particles produced in neutrino or cosmic-ray air-shower interactions. Other standard model backgrounds are considered, and found to be small.