Aims. We study the oxygen and nitrogen abundances in the interstellar medium of high-redshift galaxies. Methods. We use high resolution and high signal-to-noise ratio spectra of damped Lyman-alpha (DLA) systems detected along the line-of-sight to quasars to derive robust abundance measurements from unsaturated metal absorption lines. Results. We present results for a sample of 16 high-redshift DLAs and strong sub-DLAs (log N(H I) > 19.5, 2.4 < z(abs) < 3.6) including 13 new measurements. We find that the oxygen to iron abundance ratio is pretty much constant with [O/Fe] similar to + 0.32 +/- 0.10 for -2.5 < [O/H] < -1.0 with a small scatter around this value. The oxygen abundance follows quite well the silicon abundance within similar to 0.2 dex, although the silicon abundance could be slightly smaller for [O/H] -2. The distribution of the [N/O] abundance ratio, measured from components that are detected in both species, is somehow double peaked: five systems have [N/O] > -1 and nine systems have [N/O] < -1.15. In the diagram [N/O] versus [O/H], a loose plateau is possibly present at [N/O] similar to -0.9, which is below the so-called primary plateau as seen in local metal-poor dwarf galaxies ([N/O] in the range -0.57 to -0.74). No system is seen above this primary plateau whereas the majority of the systems lie well below with a large scatter. All this suggests a picture in which DLAs undergo successive star-bursts. During such an episode, the [N/O] ratio decreases sharply because of the rapid release of oxygen by massive stars, whereas inbetween two bursts, nitrogen is released by low and intermediate-mass stars with a delay and the [N/O] ratio increases.