There has been long-standing interest in constructing reliable memory systems from unreliable components like noisy bit-cells and noisy logic gates, under circuit complexity constraints. Prior work has focused exclusively on constructive achievability results, but here we develop converse theorems for this problem for the first time. The basic technique relies on entropy production/dissipation arguments and balances the need to dissipate entropy with the redundancy of the code employed. A bound from the entropy dissipation capability of noisy logic gates is used via a sphere-packing argument. Although a large gap remains between refined achievability results stated herein and the converse, some suggestions for ways to move forward beyond this first step are provided.