The sum-product algorithm (SPA) for the decoding of low density parity check (LDPC) codes produces exact posterior probabilities when the underlying Tanner graph is cycle-free. However, it has been shown that cycle-free Tanner graphs cannot support good codes as they have poor minimum distance properties. When a Tanner graph has cycles that do not overlap (no two cycles have any node in common), the corresponding code has managable minimal tree complexity so that optimal decoding can be achieved. In this paper, we consider codes whose Tanner graphs contain non-overlapping cycles. We derive upper bounds on the minimum distance for such codes, and show that the upper bound can be at most one more than that of a tree code for any given block length and rate. Our results imply that Tanner graphs of good codes will contain several cycles that overlap.