ABSTRACT A tight relation between the C ii 158 $\mu$m line luminosity and star formation rate is measured in local galaxies. At high redshift (z > 5), though, a much larger scatter is observed, with a considerable (15–20 per cent) fraction of the outliers being C ii-deficient. Moreover, the C ii surface brightness ($\Sigma_{\rm C\, \small {II}}$) of these sources is systematically lower than expected from the local relation. To clarify the origin of such C ii-deficiency, we have developed an analytical model that fits local C ii data and has been validated against radiative transfer simulations performed with cloudy. The model predicts an overall increase of $\Sigma_{\rm C\, \small {II}}$ with ΣSFR. However, for ΣSFR ${\gtrsim} 1 \, \mathrm{M}_\odot \,{\rm yr}^{-1}\,{\rm kpc}^{-2}$, $\Sigma_{\rm C\, \small {II}}$ saturates. We conclude that underluminous C ii systems can result from a combination of three factors: (a) large upward deviations from the Kennicutt–Schmidt relation (κs ≫ 1), parametrized by the ‘burstiness’ parameter κs; (b) low metallicity; (c) low gas density, at least for the most extreme sources (e.g. CR7). Observations of C ii emission alone cannot break the degeneracy among the above three parameters; this requires additional information coming from other emission lines (e.g. O iii88 $\mu$m, C iii1909 Å, CO lines). Simple formulae are given to interpret available data for low- and high-z galaxies.