  The recent discovery of a diffuse cosmic neutrino flux extending up to PeV
energies raises the question of which astrophysical sources generate this
signal. One class of extragalactic sources which may produce such high-energy
neutrinos are blazars. We present a likelihood analysis searching for
cumulative neutrino emission from blazars in the 2nd Fermi-LAT AGN catalogue
(2LAC) using an IceCube neutrino dataset 2009-12 which was optimised for the
detection of individual sources. In contrast to previous searches with IceCube,
the populations investigated contain up to hundreds of sources, the largest one
being the entire blazar sample in the 2LAC catalogue. No significant excess is
observed and upper limits for the cumulative flux from these populations are
obtained. These constrain the maximum contribution of the 2LAC blazars to the
observed astrophysical neutrino flux to be $27 \%$ or less between around 10
TeV and 2 PeV, assuming equipartition of flavours at Earth and a single
power-law spectrum with a spectral index of $-2.5$. We can still exclude that
the 2LAC blazars (and sub-populations) emit more than $50 \%$ of the observed
neutrinos up to a spectral index as hard as $-2.2$ in the same energy range.
Our result takes into account that the neutrino source count distribution is
unknown, and it does not assume strict proportionality of the neutrino flux to
the measured 2LAC $\gamma$-ray signal for each source. Additionally, we
constrain recent models for neutrino emission by blazars.
