It has become a commonplace that triangular systems are solved to higher accuracy than their condition would warrant. This observation is not true in general, and counterexamples are easy to construct. However, it is often true of the triangular matrices from pivoted LU or QR decompositions. It is shown that this fact is closely connected with the rank-revealing character of these decompositions. (Also cross-referenced as UMIACS-TR-95-91)