We present new data for five underluminous Type II-plateau supernovae (SNe IIP), namely SN 1999gn, SN 2002gd, SN 2003Z, SN 2004eg and SN 2006ov. This new sample of low-luminosity SNe IIP (LL SNe IIP) is analysed together with similar objects studied in the past. All of them show a flat light-curve plateau lasting about 100 d, an underluminous late-time exponential tail, intrinsic colours that are unusually red, and spectra showing prominent and narrow P Cygni lines. A velocity of the ejected material below 10(3) km s(-1) is inferred from measurements at the end of the plateau. The Ni-56 masses ejected in the explosion are very small (< 10(-2) M-circle dot). We investigate the correlations among Ni-56 mass, expansion velocity of the ejecta and absolute magnitude in the middle of the plateau, confirming the main findings of Hamuy, according to which events showing brighter plateau and larger expansion velocities are expected to produce more Ni-56. We propose that these faint objects represent the LL tail of a continuous distribution in parameters space of SNe IIP. The physical properties of the progenitors at the explosion are estimated through the hydrodynamical modelling of the observables for two representative events of this class, namely SN 2005cs and SN 2008in. We find that the majority of LL SNe IIP, and quite possibly all, originate in the core collapse of intermediate-mass stars, in the mass range 10-15 M-circle dot.