If a character fails a roll during an encounter, but has not busted the threshold level, can he attempt another roll using the same skill in that encounter?

Yes, the roll is made and the next stage of the encounter plays out based on the result. When the characters get to another point where a roll would need to be made, then the next roll is made and the result influences the next stage of the encounter. As long as the number of failed rolls doesn't exceed the tolerance rating for the encounter, the players can continue rolling for their characters.

Sorry for the slow response, but it took me awhile to find the necessary sections to refresh my memory.

Did the characters fail the Introduction Phase or Interaction Phase? Because it makes a difference.

Characters who fail at Courtesy are unlikely, to be respectfully listened to during the Interaction Phase. This means the character tripped while bowing, or used the wrong form of address, perhaps they said hello to the majordomo - not the Elven King. Some kind of social gaff that negatively impacts further events, but does not completely stop the character's participation. This does not count as a failure against the Tolerance rating of the meeting.

Whereas a character who fails during the Interaction Phase has not conveyed their issue / concern sufficiently to engage the Target's (whomever the character's are talking too) willing participation. This would be the equivalent of Thorin Oakenshield, in response to the Laketown guard's request for identification, responding with "just some dwarven travelers". This is the wrong approach and would result in being told to come back in the morning when the gates re-open. Not with the gates being thrown open and hailed as the return of the King Under the Mountain. This is a failure and does count against the Tolerance of the meeting.

As to whether a character, having failed one skill test, can make another with the same skill to the same Target, I would say yes. Note that this is my interpretation, not a rule from the books, because the rules do not actually come straight out and state this specifically, the closest they come to an answer is through implication.

None of the pages in either book, under the Encounters headings, make any mention of the actual game mechanic used for handling the dice rolls involved. The rulebooks tell you how to Introduce yourself and what skills to use; they also tell you which skills are effective to use while Interacting. But they never tell you that the Encounter rules are, to generalize quite a bit, a form of Prolonged Action but using a different game mechanic (Tolerance). So the normal ruling of not allowing the repeating of a failed roll does not apply here.

The rest is as Valarian states, with each success / failure affecting the NPC's response to the character's request.

The pertinent sections of the rules, and the subjects covered, are:

• Encounters (LMB 51-3): Discussion of Rolling vs Roleplaying an encounter, Tolerance rating (Base Rating and Modifiers - Standing, Prejudice & Additional Modifiers), Awarding AP for Encounters.
• Encounter (AB 163-5): Introduction (Useful Abilities, Suggested Skills) and Interaction (Suggested Skills).
• Prolonged Actions (LMB 22): While mechanically different, Encounters use the same basic premise, of using a skill over time to generate results, but add Tolerance.