High breakfalls are a mandatory skill. One goal of ukemi is to be able to protect yourself even if nage is trying to hurt you.
By the time a student is comfortable with normal rolls, it's appropriate to start teaching "easy" breakfalls, such as the "sliding into home" breakfall where you simply breakfall out of a normal roll. Once they can do that, you can start raising the pivot point by having them roll over a partner's hand. Start with the hand at ankle level and raise it as they become comfortable.
Failed attempts are something to avoid in any kind of ukemi. That's a good way to get hurt. If you're getting lots of failed attempts, you might step back to an easier progression (lower support point, more carefully supported falls, etc.) for a while.
Judoka learn breakfalls from the first day they step on the mat. I think the perception that they are an "advanced" skill may do more harm than good.
I don't know that they are mandatory as in you have to be able to perform a high breakfall in order to progress in the art but they are mandatory in order to protect yourself.
If a student can do a good forward roll they are ready for breakfalls. This can happen after only a few minutes of training though under strict supervision.
Lots of failed attempts means you don't have the basics down and you need to go back to square one.
Ok so Katherine covered everything, I just said the same thing slightly differently.