Quartet-based phylogeny reconstruction methods, such as Quartet Puzzling, were introduced in the hope that they might be competitive with maximum likelihood methods, without being as computationally intensive. However, despite the numerous quartet-based methods that have been developed, their performance in simulation has been disappointing. In particular, Ranwez and Gascuel, the developers of one of the best quartet methods, conjecture that quartet-based methods have inherent limitations that make them unable to produce trees as accurate as neighbor joining or maximum parsimony. In this paper, we present Short Quartet Puzzling, a new quartet-based phylogeny reconstruction algorithm, and we demonstrate the improved topological accuracy of the new method over maximum parsimony and neighbor joining, disproving the conjecture of Ranwez and Gascuel. We also show a dramatic improvement over Quartet Puzzling. Thus, while our new method is not compared to any ML method (as it is not expected to be as accurate as the best of these), this study shows that quartet methods are not as limited in performance as was previously conjectured, and opens the possibility to further improvements through new algorithmic designs.