  We investigate simulation-based bandpower covariance matrices commonly used
in cosmological parameter inferences such as the estimation of the
tensor-to-scalar ratio $r$. We find that upper limits on $r$ can be biased low
by tens of percent. The underestimation of the upper limit is most severe when
the number of simulation realizations is similar to the number of observables.
Convergence of the covariance-matrix estimation can require a number of
simulations an order of magnitude larger than the number of observables, which
could mean $\mathcal{O}(10\ 000)$ simulations. This is found to be caused by an
additional scatter in the posterior probability of $r$ due to Monte Carlo noise
in the estimated bandpower covariance matrix, in particular, by spurious
non-zero off-diagonal elements. We show that matrix conditioning can be a
viable mitigation strategy in the case that legitimate covariance assumptions
can be made.
