We present chemical abundances of carbon (C) and oxygen (O) in the Large and Small Magellanic Clouds from deep and high-quality optical spectra of H II regions. The data have been taken using the Ultraviolet-Visual Echelle Spectrograph at the 8.2-m Very Large Telescope with the goal of detecting the faint C II and O II recombination lines. For all the objects of the sample, we determine C^{2+}abundances from recombination lines and O^{2+} abundances from recombination lines and collisionally excited lines. In addition, we calculate the abundance discrepancy factors (ADFs) for O^{2+} and C^{2+}, as well as the O/H, C/H and C/O ratios. We study the behaviour of the ADF comparing the values obtained in the Magellanic Clouds with those obtained for other H_{II} regions in different galaxies. We also compare the nebular and stellar abundances in two regions of the sample. Finally, we discuss the chemical evolution of the MCs through the O/H, C/H and C/O radial gradients and the changes of the C/O ratio as a function of O/H.