We re-analyzed the carbon and oxygen isotopic ratios in the atmospheres of the two bright K giants Arcturus ({alpha} Boo) and Aldebaran ({alpha} Tau). These stars are in the evolutionary stage following the first dredge-up (FDU). Previous determinations (dating back more than 20 years) of their ^16^O/^18^O ratios showed a rough agreement with FDU expectations; however, the estimated ^16^O/^17^O and ^12^C/^13^C ratios were lower than in the canonical predictions for red giants. Today these anomalies are interpreted as signs of the occurrence of non-convective mixing episodes. We therefore re-investigated this problem to verify whether the observed data can be reproduced in this scenario and if the fairly well determined properties of the two stars can help us in fixing the uncertain parameters that characterize non-convective mixing and in constraining its physical nature.