I usually only find the word abdicate in cases where the responsibility goes a little but further than that.
Recently, the Pope abdicated, and so did the Dutch queen. I do not necessarily agree with the use of the word "fail" because the negative connotations. These people "simply" gave up their office.
This is a verb that is usually used when someone resigns from an office that is normally for life. I have never heard of a president who abdicated, but kings, queens, and (only twice in history) the pope can do that.
In many modern monarchies, abdication is a common thing, since kings an queens are no longer likely to die on the battlefield, causing them to live to ages that were unheard of when the institution was invented. Many a king or queen feels that when they become too old to fulfil their duties, instead of waiting to die, they "retire".
As to the usage, I have never heard it being used transitively - a king does not abdicate his reign, he abdicates.
OALD disagrees with me and states that "She was forced to abdicate the throne of Spain" is fine. I guess so - in cases where one has several offices, one does not necessarily abdicate all of them at once :)
It seems it is also used in a bit "looser" sense indeed of giving up responsibility, as MaulikV's example shows. It should be noted, however, that even then, we are talking about major (national) responsibilities.
I feel it would be quite overstated to use it in the context of you not wanting to repair your own TV.