Why aren't you allowed to say "I have been knowing her for ten years" or "It has been broadcasting for many years"? But you are allowed to say "I have been waiting for an hour".
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Knowing happens but once -- unlike waiting, you don't stretch the process of knowing someone over an extended period. It happens at a moment of time. That is why, I suppose.
You could, on the other hand, be getting to know the person better over ten years.
Many words describing senses and emotions tend not to be used in the progressive/continuous form. Although many such verbs are either so abstract as to make temporal distinction unnecessary, or relate to a single instance (as Kris points out with "know") it isn't the case with all of them.
Such verbs are called stative (or state) verbs and include: like, love, believe, have (when used to mean "to own") and taste.
There's a complete list here: http://www.perfect-english-grammar.com/stative-verbs.html
It’s because know describes a state of mind rather than an action, and such verbs are rarely found with a progressive construction. They are generally verbs which describe perception, emotion and thinking.
Broadcast is not a good example, because, as Hugo has explained, it can take a progressive construction.
“[Subject] has/have been [verb]ing” is a construction called the present perfect continuous.¹ It describes an ongoing or continuous action which began in the past and continued until now. As Barrie says (he calls it by another name, the progressive), this construction is typically used for actions, not states of mind.
There is only one sense of know which describes an action, not a state of mind: the archaic (Biblical) sense “to have sexual intercourse with”.² Consequently, some of your readers will wonder if you have been having sex with her for the last ten years.
This confusion is why you should probably avoid the progressive construction in that example.
Nothing wrong here, assuming you mean to describe the ongoing action of broadcasting. It should refer back to something which is capable of broadcasting continuously, such as a radio station.
Nothing wrong here either. This sentence describes the ongoing or continuous action of waiting.
Note I am now uncertain about my analysis here - I shall keep the answer up for reference sake. I expect to get to the bottom of this, for now, regard it as contrarian argumentation in the spirit of Christopher Hitchens.
Who says you aren't allowed to? It looks like Standard English to me.
It isn't really precise English because it introduces an abstraction, apparently to no end. Compare your example
Here, (1) introduces your relationship through the indirect construction of inhabiting a state while (2) simply affirms the relationship.
If you were to make use of the abstraction, this would be good English, e.g., in
"knowing" is the first thing we say about a relationship that we then elaborate on. Here, the progressive -ing form is an asset.
Some examples of "been knowing"
A quick web search shows:
These do seem to be Standard English to me. They all take plural objects, which may be a consideration, but I expect I would find examples with singular objects if I spent more time looking.
Following Hellion's doubts, I've queried the Corpus of Contemporary American English, and have seven more examples in the present continuous, consisting of quotations from newspapers and transcripts from television shows; there are certainly many more to be found:
I found no examples of this construction in journalism outside of quotes, indicating that the construction is not used in formal, edited contexts. These seem to be a wealth of idiomatic informal Standard English.
In example 5, from Jerry Springer's bastion of eloquence, the two sentences make an effective parallelism between the present continuous of knowing and dating. I contend that this is not only Standard English, but an exemplar of good, effective English.