Just being overly cautious. I wanted to avoid crossing the line from mere imitating (which I don’t seriously doubt) and comprehension, or Og forbid, telepathy, as in The N’kisi Project.
Ok, looked that up, and found various sources with various answers. Your answer is beating mine 3:2, though, so you get the point. 
Found another one: a building of rooks.
Related anecdote:
Many, many years ago my parents got a new telephone in their bedroom that had a very distinctive ringtone. It was a relatively high-pitched electronic tone and had a slow warbling rhythm (and this was back in the day when most phones still had that standard ‘metallic bell’ kind of ring). After a while, the phone developed a peculiar habit of ringing when there was no incoming call. You’d hear it ring and pick up the receiver only to be met with a dialtone. We figured it was some kind of fault in the phone, but it only occurred occasionally, so it wasn’t replaced. This went on for several weeks before we discovered the true cause. A mockingbird had taken up residence in a bush right outside my parents’ bedroom window and was imitating the ringtone!
We used to have a cockatiel (small parrot) who was bonded intensely to my wife, less so to me. When she was working late, he’d hang out with me, but he missed her. Pretty quickly he began to exactly mimick the sound of the cell phone ringing – but he would only do so when she was late and I was with him. It appeared he’d observed that the cell usually rang just before she got home. He’d walk down my arm, nudge me, and make the cell phone ring sound. I believe he was either trying to make her come home, or trying to ask me to call her.
At any rate, he never did it at other times. He wasn’t just imitating a sound, he was making the sound for a purpose, based on his observations of our behavior.
Sailboat
I’ve posted this (anecdote) before; I’ve noticed that with a couple of words and phrases that my pet budgie has learned to mimic, he actually started reproducing the sounds quite some time (weeks to months) after I stopped trying to teach them to him, and when he first tried making the sounds, he wasn’t reproducing them accurately, but with a little more practice (and significantly - no further repetition of the sounds by me), he’d get them spot on.
I think this strongly suggests that the bird has a conscious inner thought-life and is able to ‘replay’ the sound to himself, then practice against that memory (much as a human impersonator might).
I’ve also seen a lot of contextual behaviour like the stuff Sailboat mentions above - on the face of it, it’s easy to dismiss this just as conditioned reflex, but when you witness many different instances of it personally, they do add up to a very compelling picture of actual cognition - in a way that can’t always be written down adequately - exactly as it would be with trying to describe the intelligence of a human child. I’m not in any great doubt that there’s actual thinking going on in the mind of my pet bird.
I don’t make the mistake of thinking he’s a feathery little man, but he is (in my carefully-considered opinion), a person - in some quite valid sense of the word.
To add another anecdote to the pile: Most of the time when my bird has had enough play, he’ll just return to his cage and we leave him alone and/or shut him in, but if someone is trying to coax him out or play with him when he’s tired and has returned to the cage, he’ll try to shut the door himself - I can’t think of a simpler explanation than some kind of cognition for this behaviour.
egad…it looks like I’m paying the price for spending Sunday on Merrimack Road.
S.
Mangetout addressed this already regarding his budgie, but I’m curious about other species, too. Is there a learning process for all birds, or are some like “savants” who can pick up and mimic calls instantly?
I would think the Lyrebird must be able to pick up sounds almost straight away, since it’s not going to hear exactly the same version of chainsaw noise twice - unless it’s sort of learning a generic chainsaw sound and synthesising a plausible replica that doesn’t necessarily reproduce any specific incident - which in a way, would be even more clever.