  We show that, for quasi-greedy bases in Hilbert spaces, the associated
conditionality constants grow at most as $O(\log N)^{1-\epsilon}$, for some
$\epsilon>0$, answering a question by Temlyakov. We show the optimality of this
bound with an explicit construction, based on a refinement of the method of
Olevskii. This construction leads to other examples of quasi-greedy bases with
large $k_N$ in Banach spaces, which are of independent interest.
