The Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909), in Parashat Korah (9), discusses the situation of a milk or meat pot that had not been used in the past twenty-four hours (listen to audio recording for precise citation). When a person cooks meat in a pot, for example, the walls of the pot absorb the flavor of the meat, and therefore one may not cook milk in the pot, since the milk will then absorb the meat flavor from the walls. However, after twenty-four hours, the absorbed flavor becomes "Pagum," or spoiled, and loses its status as meat. Therefore, if a person mistakenly boiled milk in a meat pot that had not been used in the previous twenty-four hours, the milk is permissible for consumption. Since the meat flavor absorbed in the pot’s walls has already spoiled, it does not render the milk forbidden.
However, this Halacha applies only "Bedi’abad," meaning, after the fact. If a person mistakenly cooked milk in a meat pot, and the pot had not been used in the last twenty-four hours, then the milk is permissible. But one is not permitted to intentionally use such a pot for milk; it is only after the fact that the milk is permissible if it was mistakenly boiled in a meat pot twenty-four hours after the pot had been used.
There is another opinion, that of Rashi (Rabbi Shelomo Yishaki of Troyes, France, 1040-1105) and Rabbenu Tam (France, 12th century), who held that absorbed flavor spoils after just twelve hours. Halacha does not follow this view; the Shulhan Aruch rules that, as mentioned, absorbed flavor is considered spoiled only twenty-four hours after the pot has been used. Nevertheless, Rashi and Rabbenu Tam’s position becomes significant in a case where one is unsure how much time has passed since the pot was last used. If a person knows with certainty that a meat pot had not been used in twelve hours, but is unsure whether twenty-fours had passed, and he accidentally cooked milk in the pot, then the milk is permissible. This is a situation referred to in Halachic jargon as "Sefek Sefeka," or "double doubt." First, it is possible that twenty-four hours had passed since the pot had been used with meat, in which case the milk is permissible. And even if only twelve hours have passed, there is the possibility that Rashi and Rabbenu Tam were correct, that twelve hours suffices to render absorbed taste spoiled. In such a case, then, it would be permissible to drink the milk cooked in the meat pot.
Summary: It is forbidden to cook milk in a pot that is used with meat (or vice versa). If one accidentally cooked milk in a meat pot, the milk may be eaten if the pot had not been used for twenty-four hours before the milk was cooked. Additionally, if one is unsure whether the pot had been used in the previous twenty-four hours, but he knows for certain that it had not been used in the previous twelve hours, then the milk is permissible. In all other instances, however, the milk is forbidden.
