Turbo Codes and multiple parallel concatenated codes (MPCCs) yield performance very close to the Shannon limit. However, they are not asymptotically good, in the sense of having the minimum distance grow linearly with the length of the code. At the other extreme, multiple serially concatenated codes (MSCCs), for example very simple repeat-accumulate-accumulate codes, have proven to be asymptotically good, but they suffer from a convergence threshold far from capacity. In this paper, we investigate hybrid concatenated coding structures consisting of an outer MPCC with very simple memory-1 component encoders serially concatenated with an inner accumulator. We show that such structures exhibit linear distance growth with block length and that they have better thresholds than MSCCs. The results indicate a fundamental tradeoff between minimum distance growth and convergence threshold in turbo-like codes.