We have obtained near-infrared observations of some of the faintest objects so far known towards the Pleiades young stellar cluster, with the purpose of investigating the sequence that connects cluster very low-mass stars with substellar objects. We find that infrared data combined with optical magnitudes are a useful tool to discriminate cluster members from foreground and background late-type field stars contaminating optical surveys. The bottom of the Pleiades sequence is clearly defined by the faint HHJ objects as the very low-mass stars approaching the substellar limit, by the transition object PPl 15, which will barely ignite its hydrogen content, and by the two brown dwarfs Calar 3 and Teide 1. Binarity amongst cluster members could account for the large dispersion observed in the faint end of the infrared colour-magnitude diagrams. Two objects in our sample, namely HHJ 6 and PPl 15, are overluminous compared to other members, suggesting a probable binary nature. We have reproduced the photometric measurements of both of them by combining the magnitudes of cluster very low-mass stars and brown dwarfs and using the most recent theoretical evolutionary tracks. The likely masses of the components are slightly above the substellar limit for HHJ 6, while they are 0.080 and 0.045+/-0.010Msun_ for PPl 15. These masses are consistent with the constraints imposed by the published lithium observations of these Pleiads. We find a single object infrared sequence in the Pleiades cluster connecting very low-mass stars and brown dwarfs. We propose that the substellar mass limit (~0.075Msun_) in the Pleiades (~120Myr) takes place at absolute magnitudes M_I_=12.4, M_J_=10.1, M_H_=9.4 and M_K_=9.0 (spectral type M7). Cluster members fainter by 0.2mag in the I-band and 0.1mag in the K-band should be proper brown dwarfs. The star-brown dwarf frontier in the Hyades cluster (600Myr) would be located at M_I_=15.0, M_J_=11.6, M_H_=10.8 and M_K_=10.4 (spectral type around M9). For an age older than 1000Myr we estimate that brown dwarfs are fainter than M_K_=10.9 (spectral type later than M9.5).