We propose a four-neutrino model which can reconcile the existing data coming from underground experiments in terms of neutrino oscillations, together with the hint from the LSND experiment and a possible neutrino contribution to the hot dark matter of the Universe. It applies the idea that extra compact dimensions, probed only by gravity and possibly gauge-singlet fields, can lower the fundamental scales such as the Planck, string or unification scales. Our fourth light neutrino νs (s for sterile) is identified with the zero mode of the Kaluza-Klein states. To first approximation \nu_sterile combines with the nu_mu in order to form a Dirac neutrino with mass in the eV range leaving the other two neutrinos massless. The smallness of this mass scale (suitable for LSND and Hot Dark Matter) arises without appealing neither to a see-saw mechanism nor to a radiative mechanism, but from the volume factor associated with the canonical normalization of the wave-function of the bulk field in the compactified dimensions. % On the other hand the splitting between \nm and \nu_sterile (atmospheric scale) as well as the mass of the two other neutrinos (solar mass scale) arise from the violation of the fermion number on distant branes. We also discuss alternative scenarios involving flavour-changing interactions. In one of them \ne can be in the electron-volt range and therefore be probed in beta decay studies.