  In recent years a correlation between mass accretion rates onto new-born
stars and their proto-planetary disc masses was detected in nearby young
star-forming regions. Although such a correlation can be interpreted as due to
viscous-diffusion processes in the disc, highly-accreting sources with low disc
masses in more evolved regions remain puzzling. In this paper, we hypothesise
that the presence of a stellar companion truncating the disc can explain these
outliers. Firstly, we searched the literature for information on stellar
multiplicity in Lupus, Chamaeleon~I and Upper Sco, finding that roughly 20 per
cent of the discs involved in the correlation are in binaries or higher-order
multiple stellar systems. We prove with high statistical significance that at
any disc mass these sources have systematically higher accretion rates than
those in single-stars, with the bulk of the binary population being clustered
around $M_\mathrm{disc}/\dot{M}_\mathrm{acc}\approx0.1\,\mathrm{Myr}$. We then
run coupled gas and dust one-dimensional evolutionary models of tidally
truncated discs to be compared with the data. We find that these models are
able to reproduce well most of the population of observed discs in Lupus and
Upper Sco, even though the unknown eccentricity of each binary prevents an
object by object comparison. In the latter region, the agreement improves if
the grain coagulation efficiency is reduced, as may be expected in discs around
close binaries. Finally, we mention that thermal winds and sub-structures can
be important in explaining few outlying sources.
