I think there are several. In the Hobbit a great example is when Thorin & Company meet the Elvenking, and Thorin speaks for them and then has to decide how to proceed when he is questioned concerning his business in the forest. One might interpret that he failed that Interaction (but maybe not as much as he could have). Earlier there is the Encounter with the Great Goblin.
Famously of course, and there are rules in the Erebor supplement that mirror it more closely, there is Bilbo's altercation with Smaug.
In Lord of the Rings a great example I can think of is when Gandalf and Pippin talk to Denethor on their first arrival.
I would say that Denethor values Valour above Wisdom (his preference in his children being any indication). He's a very wise man himself and proud, and doesn't like any rivals. Gandalf and Pippin clearly had an agreement that Gandalf would handle introductions, but that Denethor would be interacting primarily with Pippin
'He will speak most to you, and question you much, since you can tell him of his son Boromir.'
This is Gandalf rolling for Insight and gaining Bonus Success Dice, possibly giving them to Pippin.
The Introduction
Gandalf probably made a Courtesy roll with his introduction, but he's not the spokesman because he knows Denethor will prefer to talk to Pippin, so they have individual introductions here.
'Hail, Lord and Steward of Minas Tirith, Denethor son of Ecthelion! I am come with counsel and tidings in this dark hour.'
Denethor seems receptive, and asks about the nature of Pippin, immediately drawing the matter to his son. Gandalf had earlier warned Pippin not to mention Frodo's errand, or Aragorn.
"It has been told to me that you bring with you one who saw my son die. Is this he?"
Gandalf answers by rolling for Riddle (perhaps successfully), not revealing too much,
'It is ... One of the twain. The other is with Theoden of Rohan and may come hereafter. Halflings they are, as you see, yet this is not he of whom the omens spoke.'
Then Denethor complains about the omen and speaks bitterly about wishing Faramir had gone instead of Boromir. Gandalf rolls for Persuade (probably not successfully),
'He would have gone, ... Be not unjust in your grief! Boromir claimed the errand and would not suffer any other to have it ... But you speak of his death. You have had news of that ere we came?'
Denethor then reveals the horn of Boromir. He speaks with "scorn and suspicion" when Pippin recognizes the horn. Pippin is stung and being filled with pride offers his services to Lord Denethor - probably Courtesy. This is Pippin's Introduction attempt.
'Little service, no doubt, will so great a lord of Men think to find in a hobbit, a halfling from the northern Shire; yet such as it is, I will offer it, in payment of my debt.' Twitching aside his grey cloak, Pippin drew forth his small sword and laid it at Denethor's feet.
This seems to be a success, and one wonders if Pippin Invoked a Trait or used a point of Hope for the purpose, or perhaps he called in that Success Dice from earlier, because
A pale smile, like a gleam of cold sun on a winter's evening, passed over the old man's face; but he bent his head and held out his hand, laying the shards of the horn aside ... 'I see strange tales are woven about you ... and once again it is shown that looks may belie the man - or the halfling. I accept your service. For you are not daunted by words; and you have courteous speech, strange though the sound of it may be to us in the South. And we shall have need of all of folk of courtesy, be they great or small, in the days to come.'
Denethor really likes Courtesy!
Now we are into Interaction.
Pippin now proceeds to swear fealty to Denethor (no rolls). Denethor then asks Pippin
'Tell me your full tale, and see that you recall all that you can of Boromir, my son. Sit now and begin!'
Gandalf becomes annoyed at this, because he has news to tell Denethor and he doesn't like being delayed, rolling for Persuade again (or Awe?),
'Is it naught to you that Theoden has fought a great battle, and that Isengard is overthrown, and that I have broken the staff of Saruman?'
Denethor is unimpressed. The roll failed. Perhaps he is prejudiced against Wizards.
' ... I know already sufficient of these deeds for my own counsel against the menace of the East.'
They then have a stare off, perhaps rolling Awe against Awe, but Pippin rolls for Insight about the two men, observing them successfully,
Denethor looked indeed much more like a great wizard than Gandalf did, more kingly, beautiful, and powerful; and older. Yet by a sense other than sight Pippin perceived that Gandalf had the greater power and the deeper wisdom, and a majesty that was veiled. And he was older, far older.
Pippin then rolls for Lore with regard to Gandalf and wizards:
'How much older?' he wondered, and then he thought how odd it was that he had never thought about it before. Treebeard had said something about wizards, but even then he not thought of Gandalf as one of them. What was Gandalf? In what far time and place did he come into the worldk, and when would he leave it?
But the Lore roll is not a success.
And then his musings broke off, and he saw that Denethor and Gandalf still looked each other in the eye, as if reading the other's mind
At this point, Denethor loses the Awe vs. Awe contest,
But it was Denethor who first withdrew his gaze.
Then Pippin has to tell his story, and do it without revealing too much. He is probably allowed to roll either Song or Riddle. He seems to choose Riddle, as he hasn't quite put it into verse. It seemed to be difficult rolls, and he probably dug into his Hope reserve more than once ...
Pippin never forgot that hour in the great hall under the piercing eye of the Lord of Gondor, stabbed ever and anon by his shrewd questions, and all the while conscious of Gandalf at his side, watching and listening, and (so Pippin felt) holding in check a rising wrath and impatience.
The Encounter is over after this and Denethor dismissed the two. Pippin fared better than Gandalf most likely, and the Encounter overall was a success, but not as successful as Gandalf hoped. Pippin has been sworn into Denethor's service and good graces, although he may have revealed more than intended in the last hour when he rolled that Eye of Sauron, and it will come back to haunt him later in the adventure, when Denethor rather loses his mind and points out the plot he thinks the wizard has. Gandalf was unable to communicate the information he wanted to, and was surprised to learn Denethor already knew a good deal of it, and refused his counsel. They part in anger. It is safe to say that there were some objectives that were not obtained.