We consider the On-Line Dual Bin Packing problem where we have a fixed number n of bins of equal size and a sequence of items. The goal is to maximize the number of items that are packed in the bins by an on-line algorithm. An investigation of First-Fit and an algorithm called Log shows that, in the special case where all sequences can be completely packed by an optimal off-line algorithm, First-Fit has a constant competitive ratio, but Log does not. In contrast, if there is no restriction on the input sequences, Log is exponentially better than First-Fit. This is the first separation of this sort with a difference of more than a constant factor. We also design randomized and deterministic algorithms for which the competitive ratio is constant on sequences which the optimal off-line algorithm can pack using at most alpha n bins, if alpha is constant and known to the algorithm in advance.