Single- and double-electron-transfer cross sections have been measured for ${\mathrm{B}}^{3+}$, ${\mathrm{C}}^{4+}$, ${\mathrm{N}}^{5+}$, and ${\mathrm{O}}^{6+}$ in collision with He at velocities between 0.5 and 1.2\ifmmode\times\else\texttimes\fi{}${10}^{8}$ cm/sec. The single-electron-capture cross sections peak near 15\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}16}$ ${\mathrm{cm}}^{2}$ for each case except ${\mathrm{C}}^{4+}$, where single capture is anomalously low. The double-capture cross sections are about 3\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}16}$ ${\mathrm{cm}}^{2}$, except for ${\mathrm{B}}^{3+}$, where the highest observed value is 1.5\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}16}$ ${\mathrm{cm}}^{2}$. These measurements for ${\mathrm{B}}^{3+}$ and ${\mathrm{C}}^{4+}$ compare well with existing experiments and theory, except for ${\mathrm{C}}^{4+}$ single capture. Within the narrow range tested, ${\mathrm{C}}^{4+}$ and ${\mathrm{B}}^{3+}$ cross sections exhibit variation with velocity, but ${\mathrm{N}}^{5+}$ and ${\mathrm{O}}^{6+}$ cross sections remain constant.