  Explaining the observed number counts of submillimetre (submm) galaxies
(SMGs) has been a longstanding challenge for theoretical models. Surprisingly,
recent observations have suggested that the brightest SMGs are almost
exclusively multiple fainter sources blended into a single source in the
single-dish surveys. This result is in contrast with the predictions of our
previously presented theoretical model, which includes some effects of
blending. In this Letter, we consider the implications of an upper limit on the
submm flux density for the demographics of the SMG population. Using a relation
amongst submm flux, star formation rate (SFR), and dust mass (M_d) from our
previous work, we infer the maximum SFR for a range of flux density limits. For
M_d = 2 x 10^9 (5 x 10^8) M_sun, the SFR limit that corresponds to an
870-{\mu}m flux density (S_870) limit in the range 9 - 12.5 mJy is in the range
~630 - 1400 (3600 - 7700) M_sun/yr. The SFR limit implies a correspondingly
sharp, redshift-dependent cutoff in the stellar mass (M_star) function, the
value of which we predict using the S_870-M_star relation predicted by our
model. The M_star limit decreases with increasing redshift: for an S_870 limit
of 9 - 12.5 mJy, the M_star limit ranges from ~4 - 7 x 10^12 M_sun at z ~ 1 to
~3 - 5 x 10^11 M_sun at z ~ 6. We discuss the few interferometrically detected
SMGs that may be brighter than the proposed cutoff. Although such objects are
certainly interesting, inferences based on such objects may not apply to most
SMGs.
