I have always said budgies are just as smart as larger parrots, and can be just (or even more) trainable than larger parrots, depending on your bird's personality. However, many people seem to be of the opinion that budgies do not even make great "pets", let alone being just as smart as their larger counterparts.
I recently heard Dr. Irene Pepperburg speak about Alex and her studies, and was thrilled to hear her opinion on the trainability and intelligence of not only other Africans Greys, but also other parrot species. To sum up: Alex was likely not an "unusual genius" but rather, an African Grey that loved learning through speech, and that had 8 hours of training a day (which makes a huuuge difference, obviously!). She went on to say how the other two greys they are working with are just as smart, but with different learning styles, so they therefore need different teaching styles and can learn different things from Alex. She also said that other species were just as smart as Greys, but again, would probably have to be taught in a different manner that worked with their particular personalities. And finally, she mentioned that though they had tried working with budgies, the shorter attention spans (and life spans) were more difficult to work with in her studies. I can't say I really blame her on that score, but I was very happy to hear how much she did love budgies, and that though they were not right for her studies, she does not feel them to be unintelligent.
However, a shorter attention span does not mean they can not be trained. I recently saw this video, and thought it portrayed exactly how amazing budgies are! Aren't they adorable?