In English having a "bird brain" is not a compliment. Bird brains are small and since the usual understanding is that size matters having a small brain is not good.
As a matter of fact scientists have been puzzled by noticing that some birds, in spite of their small brains, can perform activities that require foresight, planning and in general indicate some smartness.
Macaws can repeat what you are saying, it is not the same as talking but it is something... They also show some intelligent behaviour that does not fit with the small size of their brain.
Now in an article published on PNAS a neuroscientist at Vanderbilt University along with colleagues at the Charles University in Prague report the discovery of a probable reason for this smartness.
Birds brains, they examined some 20 species of birds showing cognitive capabilities and found that although the volume of their brains is smaller than the ones of primates, and other mammals, they contain more neurones per unit of volume. Their neurones are smaller, hence they can pack more of them.
Additionally, they pack more neurones in the frontal part of the brain, the part that is responsible to support cognitive functions.
In the end it turns out that "the more is better" still applies.