The free streaming of warm dark matter particles dampens the fluctuation spectrum, flattens the mass function of haloes and sets a fine-grained phase density limit for dark matter structures. The phase-space density limit is expected to imprint a constant-density core at the halo centre in contrast to what happens for cold dark matter. We explore these effects using high-resolution simulations of structure formation in different warm dark matter scenarios. We find that the size of the core we obtain in simulated haloes is in good agreement with theoretical expectations based on Liouville's theorem. However, our simulations show that in order to create a significant core (? kpc) in a dwarf galaxy (M˜ 1010 Msun), a thermal candidate with mass as low as 0.1 keV is required. This would fully prevent the formation of the dwarf galaxy in the first place. For candidates satisfying large-scale structure constraints (mν larger than ≈1-2 keV), the expected size of the core is of the order of 10 (20) pc for a dark matter halo with a mass of 1010 (108) Msun. We conclude that 'standard' warm dark matter is not a viable solution for explaining the presence of cored density profiles in low-mass galaxies.