We investigate whether a Gaussian likelihood, as routinely assumed in the analysis of cosmological
data, is supported by simulated survey data. We define test statistics, based on a
novel method that first destroys Gaussian correlations in a data set, and then measures the nonGaussian
correlations that remain. This procedure flags pairs of data points that depend on each
other in a non-Gaussian fashion, and thereby identifies where the assumption of a Gaussian
likelihood breaks down. Using this diagnosis, we find that non-Gaussian correlations in the
CFHTLenS cosmic shear correlation functions are significant. With a simple exclusion of the
most contaminated data points, the posterior for s8 is shifted without broadening, but we find
no significant reduction in the tension with s8 derived from Planck cosmic microwave background
data. However, we also show that the one-point distributions of the correlation statistics
are noticeably skewed, such that sound weak-lensing data sets are intrinsically likely to lead
to a systematically low lensing amplitude being inferred. The detected non-Gaussianities get
larger with increasing angular scale such that for future wide-angle surveys such as Euclid
or LSST, with their very small statistical errors, the large-scale modes are expected to be
increasingly affected. The shifts in posteriors may then not be negligible and we recommend
that these diagnostic tests be run as part of future analyses.