In “How come parrots, etc., talk, but chickens don’t” the answer was basically “we really don’t know, but we have some theories.”
Okay. Anyone have any theories as to when this ability to mimic sounds began? I begin to wonder whether T. rex could pretend to be another creature and call prey to himself…
“That’s the largest chicken I’ve ever heard!”
How does it benefit birds to be able to mimic sounds?
Considering all birds probably stem from a common theropoda ancestor, the fact that not all birds can mimic means it was something that started after birds split off from other theropoda. Swifts, Tree Swifts, and Hummingbirds all share a common ancestor and are Apodiformes. Apodiforms have a common ancestor with Passeriformes which includes all song birds. It’s likely that mimicking had developed with this ancestor.
It is possible that mimicking could have arisen through convergent evolution where traits that seem in common between two species have nothing to do with each other. Still, I’d place my money against T-Rex’s ability to ask for a cracker. If a T-Rex wanted a cracker, I’d think it could procure one without having to ask for it.
As it turns out, two of the three groups that have vocal learning in birds, the parrots and songbirds, are rather closely related and possibly their common ancestor had this trait. (However, there are closer relatives of songbirds, known as suboscine passerines, that lack this ability, and thus if this is true they must have lost it.)
However, hummingbirds are quite distantly related to parrots and passerines. Since all of the closer relatives to those groups lack vocal learning, I think it’s more likely that vocal learning is convergent between hummingbirds and parrots/songbirds.
Since most modern birds are vocal, and so are crocodilians, it’s likely that other descendants of their common ancestor such as dinosaurs also were. However, if theropods were able to mimic it would have been convergent to the ability found in birds.
The last I heard, there was a possibility that all living birds descend from a single species that survived the KT event. Things are always changing nowadays, of course…
Think of evolution in stages, not as a moment. The ability to make a range of sounds is not the ability to mimic. The ability to mimic just some sounds is not the ability to mimic many sounds. The application of mimicry is not the same as the ability. These things require a fair bit of genetic prep.
So the genes that promote mimicry no doubt developed earlier in the bird or dino lineage than mimicry itself. In that case it oversimplifies matters somewhat to speak of a straight choice between common inheritance and convergent evolution - a number of alleles are likely hanging around in the common gene pool to make that “convergent” evolution of mimicry likelier, and these were not well developed in lineages that didn’t have much use for them.
So, I don’t know when mimicry began, but it only evolved in different bird clades because the elements are in that gene pool. Just a pet theory but it seems very logical. 
So that’s a “probably?” Because I can totally see a T Rex luring its prey with a fake call, eating it, and saying, “Dumb ass,” based on how evil Talking Bird, my Cockatiel, is.
So, could T Rex mimic a female brontosaur in heat?
That would have been very useful.