We investigate the possible presence of neutral hydrogen (H I) in intergalactic filaments at very low redshift (z similar to 0.08), by stacking a set of 274 712 2dFGRS galaxy pairs over 21-cm maps obtained with dedicated observations conducted with the Parkes radio telescope, over a total sky area of approximately 1300 deg(2) covering two patches in the northern and in the southern Galactic hemispheres. The stacking is performed by combining local maps in which each pair is brought to a common reference frame; the resulting signal from the edge galaxies is then removed to extract the filament residual emission. We repeat the analysis on maps cleaned removing either 10 or 20 foreground modes in a principal component analysis. Our study does not reveal any clear H I excess in the considered filaments in either case; we determine upper limits on the total filament H I brightness temperature at T-b less than or similar to 10.3 mu K for the 10-mode and at T-b less than or similar to 4.8 mu K for the 20-mode removed maps at the 95 per cent confidence level. These estimates translate into upper limits for the local filament H I density parameter, Omega((f))(HI) less than or similar to 7.0 x 10(-5) and Omega((f))(HI) less than or similar to 3.2 x 10(-5), respectively, and for the H I column density, N-HI less than or similar to 4.6 x 10(15) cm(-2) and N-HI less than or similar to 2.1 x 10(15) cm(-2), respectively. These column density constraints are consistent with previous detections of H I in the warm-hot intergalactic medium obtained observing broad Ly a absorption systems. This work shows for the first time how such constraints can be achieved using the stacking of galaxy pairs on 21-cm maps.