Recently, secondary magnetic island (plasmoid) formation due to the tearing mode instability of the Sweet-Parker current sheet was observed in numerical simulations and was suggested as a potential mechanism for increased reconnection rates. These secondary islands were observed when the Lundquist Number S is greater than a specified threshold. Our simulations on a simple representative geometry (Magnetic Island Coalescence Instability) using a well-tested pseudo-spectral code, however, show that there is no secondary island formation or increased reconnection rate if the runs are well-resolved. The reconnection rate we obtained with such simulations are in accordance with the Sweet-Parker theory (slow reconnection model: Reconnection Rate proportional to S^1/2) which is a motivation for high spatial and numerical accuracies needed in such problems. When we add random noise throughout the simulation (supported in the recent literature), we do observe secondary island formation and increased reconnection rates that show a very weak dependence on the Lundquist Number. Some of these results will be presented in this talk.