In How I Met Your Mother Barney loses the Duckie Tie bet in October 2011 when he cannot perform the Benihana routine satisfactorily. He was to wear it for 1 year, hence October 2012. However, I was catching reruns of the episodes and saw the Broath episode, where he was not wearing it, though it would have been well inside the duckie tie year. There are two possible causes: Broath was one of Ted's memories outside the duckie tie year, and just broadcast here OR the bet was invalidated some time later. Anyone know?
There's an episode earlier in that season (before the "Broath" episode) where Barney is going to have dinner with Nora's parents, but he doesn't want to wear the tie. He exchanges 3 slaps to get rid of the tie. The episode is called "Disaster Averted".
This is a real late answer, but the broath is about Quinn, which is the season after Nora. In Disaster Averted they make a deal about the Ducky Tie where he trades more slaps to take it off before meeting Nora’s parents.