The Rabbis used to speak highly to R. Zera of the son of R. Zebid25 the brother of R. Simeon son of R. Zebid as being a great man and well versed in the benedictions. He said to them: When you get hold of him bring him to me. Once he came to his house and they brought him a loaf, over which he pronounced the blessing mozi. Said R. Zebid: Is this the man of whom they say that he is a great man and well versed in benedictions? Had he said ha-mozi, he would have taught us the meaning of a text and he would have taught us that the halachah is as stated by the Rabbis. But when he says mozi, what does he teach us?1 In fact he acted thus so as to keep clear of controversy. And the law is that we say, ha-mozi bread from the earth', since we hold with the Rabbis who say that it means 'who has brought forth'.Thus, someone who sought to fulfill according to all positions, and thus keep clear of controversy, was roundly criticized as a know-nothing, since in doing so, his actions do not teach that we hold like the Rabbis (Hamotzi) over Rabbi Nechemia (Motzi). Because if one said Motzi, everyone would agree you fulfilled, yet he was criticized for it.
And so is the halacha. And so every day, when we make a beracha on bread, of specifically HaMotzi, we put this idea into action, and perhaps internalize the idea that sometimes, we pasken like X over Y.
However, on 39b, we read the following:
It has been stated: If pieces and whole loaves are set before one, R. Huna says that the benediction can be said over the pieces,2 and this serves also for the whole loaves, whereas R. Johanan says that the religious duty is better performed if the blessing is said over the whole one. If, however, a broken piece of wheat bread and a whole loaf of barley bread are set before one, all agree that the benediction is said over the piece of wheaten bread, and this serves also for the whole loaf of barley bread. R. Jeremiah b. Abba said: There is the same difference of opinion between Tannaim:3 Terumah is given from a small whole onion, but not from the half of a large onion. R. Judah says: Not so, but also from the half of a large onion.4 Are we to say that the point in which they differ is this: one authority holds that the fact of being worth more is more important, while the other holds that the fact of being whole is more important? — Where a priest is on the spot,5 all agree that the fact of being worth more is more important. Where they differ is when there is no priest on the spot, since we have learnt: Wherever a priest is on the spot,terumah is given from the best of the produce; where the priest is not on the spot,6 terumah is set aside from that which will keep best. R. Judah said: Terumah is in all cases given from the best.7 R. Nahman b. Isaac said: A Godfearing man will seek to satisfy both.8 Who is such a one? Mar the son of Rabina. For Mar the son of Rabina used to put the broken piece under9 the whole loaf and then break the bread.10 A Tanna recited in the presence of R. Nahman b. Isaac: One should place the broken piece under the whole loaf and then break and say the benediction. He said to him: What is your name? Shalman, he replied. He said to him: Thou art peace [shalom] and thy Mishnah is faultless [shelemah], for thou hast made peace between the scholars.It is unclear whether Rav Huna was saying that the bracha can be said over the pieces or that it was optimal to say it over the pieces. (Rashi adds that the pieces were bigger, thus giving a reason for precedence for the pieces.) Yet here, Mar bar Ravina would satisfy both positions, and was called a God-fearing man for it. And Rav Nachman bar Yitzchak advocated such an approach. And indeed, a reciter of Tannaitic statements recited before the same Rav Nachman bar Yitzchak, advocating such an approach, and was praised for it.
What distinguishes these cases? Perhaps there was no clear decision? Perhaps it is a difference in the approach of different Amoraim? I would guess that the reason to try to fulfill both specifically here is that there are different, conflicting versions of the optimum, and by doing this, one gives "precedence" to both.
To see this, we should look to the other case of Mar b'reih d'Ravina, and Rav Nachman bar Yitzchak, in Shabbat 61a:
For R. Johanan said: Like tefillin, so are shoes: just as tefillin [are donned] on the left [hand], so are shoes [put on] the left [foot first]. An objection is raised: When one puts on his shoes, he must put on the right first and then the left? — Said R. Joseph: Now that it was taught thus, while R. Johanan said the reverse, he who acts in either way acts [well].4 Said Abaye to him: But perhaps R. Johanan did not hear this Baraitha, but if he had heard it, he would have retracted? Or perhaps he heard it and held that the halachah is not as that Mishnah?5 R. Nahman b. Isaac said: A God-fearing person satisfies both views. And who is that? Mar, the son of Rabina. What did he do? He put on the right foot [sandal] but did not tie it. Then he put on the left, tied it, and then tied the right [sandal]. R. Ashi said: I saw that R. Kahana was not particular.Perhaps it is something particular to Rav Nachman bar Yitzchak, who explicitly sought out the trait of being a yerei Shamayim, one who had fear of Heaven? Consider Shabbat 156b:
From R. Nahman b. Isaac too [we learn that] Israel is free from planetary influence. For R. Nahman b. Isaac's mother was told by astrologers, Your son will be a thief. [So] she did not let him [be] bareheaded, saying to him, 'Cover your head so that the fear of heaven may be upon you, and pray [for mercy]'. Now, he did not know why she spoke that to him. One day he was sitting and studying under a palm tree; temptation13 overcame him, he climbed up and bit off a cluster [of dates] with his teeth.14Meanwhile, this same Rav Nachman bar Yitzchak was the one who advised avoiding following the position of Bet Shammai. In Berachot 11a:
R. Nahman b. Isaac said: One who follows the rule of Beth Shammai makes his life forfeit, as we have learnt: R. TARFON SAID: I WAS ONCE WALKING BY THE WAY AND I RECLINED TO RECITE THE SHEMA' IN THE MANNER PRESCRIBED BY BETH SHAMMAI, AND I INCURRED DANGER FROM ROBBERS. THEY SAID TO HIM: YOU DESERVED TO COME TO HARM, BECAUSE YOU ACTED AGAINST THE OPINION OF BETH HILLEL.Ignoring my diyuk elsewhere that he was actually losing fulfillment of uvelechtecha vadarech according to Bet Hillel, we see here that Rabbi Tarfon was actually fulfilling according to both Bet Shammai and Bet Hillel.
But, there was no specific kiyum according to Bet Hillel, and his act was a demonstrative act against the lenient position of Bet Hillel. So maybe this is not "fulfilling both of them".
What about in general? IIRC, the general pattern of the Mishna Berura (unlike certain other Acharonim) is to satisfy multiple shitot simultaneously. Not everyone holds like the Mishna Berura in this.
What about, e.g., in tznius. We hold that sheitels are OK, and that shok is the thigh. Vocal people who hold otherwise try to persuade us by saying, if great rabbi X, Y, and Z said it was assur, wouldn't you avoid possibly violating, based on his say-so? And are you not a callous sinner if you disregard his words? The answer is no, there is a pesak and we hold like our pesak.