  We consider the classic joint source-channel coding problem of transmitting a
memoryless source over a memoryless channel. The focus of this work is on the
long-standing open problem of finding the rate of convergence of the smallest
attainable expected distortion to its asymptotic value, as a function of
blocklength $n$. Our main result is that in general the convergence rate is not
faster than $n^{-1/2}$. In particular, we show that for the problem of
transmitting i.i.d uniform bits over a binary symmetric channels with Hamming
distortion, the smallest attainable distortion (bit error rate) is at least
$\Omega(n^{-1/2})$ above the asymptotic value, if the ``bandwidth expansion
ratio'' is above $1$.
