We are up to Mishnah 7 in the fifteenth chapter of Maseches Yevamos. The Mishnah presents three cases of doubt, and in all of them Rabbi Tarfon and Rabbi Akiva disagree.
The first case - "Amrah meis ba'ali ve'achar kach meis chami":
A woman who says that her husband died and that afterward her father-in-law died - the halachah is "hi noseses venoteles kesubasah": she herself is permitted to marry, and she collects her kesubah on the strength of her own testimony. "Vachamosah asurah" - but her mother-in-law remains forbidden. The reason, as we learned in Mishnah 4: just as the mother-in-law is not believed to testify to the death of her son in order to permit her daughter-in-law to marry, so too the daughter-in-law is not believed to testify that her father-in-law has died.
And even though according to the daughter-in-law's own testimony, at the time her father-in-law died he was no longer her father-in-law, and that woman is no longer her mother-in-law either - since her husband had already died beforehand - nevertheless, while she is believed with regard to her own husband, we do not make use of that credibility in order to benefit her mother-in-law, and therefore the mother-in-law remains forbidden.
Do we then treat the mother-in-law as one who is certainly still married?
On this the Mishnah rules: "Chozeres bas Yisrael lakohen - tochal biTerumah, divrei Rabbi Tarfon". The case is a mother-in-law who is a bas Yisrael, who originally did not eat Terumah, and only because of her marriage to a kohen was she permitted to do so. Rabbi Tarfon holds that she continues to eat Terumah, in line with his position in the previous Mishnah: we assume that the husband is still alive, to the point that we permit her even to eat Terumah.
Rabbi Akiva, however, repeats his position from the previous Mishnah: "Ein zu derech motzi'asah midei aveirah, ad shetehei asurah lehinasei va'asurah le'echol biTerumah" - this is not the way to keep her from transgression. Since we do not believe this testimony to permit the mother-in-law to marry, because of that very same doubt - perhaps the daughter-in-law is telling the truth and the mother-in-law's husband is no longer alive - she is also forbidden to eat Terumah.
The second case - "Kidesh achas mechamesh nashim":
A man who was mekadesh one of five women, "ve'ein yodei'a eizeh kidesh", "kol achas omeres osi kidesh". In Rabbi Tarfon's view: "nosen get lechol achas ve'achas" - he must give a get to each one of them in order to permit them, since each one of them is a possible eishes ish. "Umani'ach kesubah beineihen umistalek" - he places among them the payment of the kesubah that he obligated himself to give the woman he was mekadesh, and he goes on his way. At that point his responsibility ends: whichever one is the actual woman will take the kesubah, and they will work it out among themselves as they see fit. "Divrei Rabbi Tarfon".
Rabbi Akiva says: "Ein zu derech motzi'aso midei aveirah, ad sheyiten get uchesubah lechol achas ve'achas" - placing the kesubah among them does not discharge his obligation; rather, he must give a get and a kesubah to each and every one of the women.
Some explain Rabbi Tarfon's position this way: ordinarily a woman who is only betrothed has no kesubah, and the case here is one in which he wrote a kesubah for one of them, and she lost her kesubah. That is why each one of them claims, "I am the one, and I am the one who lost the kesubah" - and therefore he bears no further obligation.
As for Rabbi Akiva's position, obligating him in a kesubah to each and every one, for otherwise he has not removed himself from transgression: the Gemara establishes this in a very specific case - that he performed the kiddushin through bi'ah. Even though by Torah law bi'ah is one of the means of kiddushin, the Rabbis held that one should not act this way; and since he transgressed a rabbinic prohibition, we are stringent with him and obligate him in a kesubah to each of them. But had he been mekadesh with money or with a document, even Rabbi Akiva would agree that it is enough to place the kesubah among them: whoever brings proof that she is the woman will take it, and if not, they must find a solution on their own.
The third case - "Gazal echad mechamishah":
One who stole from one of five people, "ve'ein yodei'a eizeh nigzal", "kol echad omer osi gazal". In Rabbi Tarfon's view: "mani'ach gezeilah beineihen umistalek" - he places the stolen object among them, and with that he has fulfilled his obligation to return it and he goes on his way. Rabbi Akiva says: "Ein zu derech motzi'aso midei aveirah, ad sheyeshalem gezeilah lechol echad ve'echad" - he must pay the value of the theft to each and every one of the possible victims.
In summary: in this Mishnah we learned three cases - a woman who testified "my husband died and afterward my father-in-law died" and the status of her mother-in-law regarding Terumah, one who was mekadesh one of five women and does not know which of them, and one who stole from one of five people and does not know from whom he stole. In all of them Rabbi Tarfon holds that it is enough to place the money among them and withdraw, while Rabbi Akiva requires him to satisfy every single doubt, for otherwise "ein zu derech motzi'aso midei aveirah".