Can animals speak like humans?
It depends on what is understood by speaking.
Anatomy  If by speaking one means producing sounds similar to those produced by humans, than the ability to speak is largely determined by the anatomy of the speech organs - whether they can produce the same type of sound as human does. There are very few animals capable of this (I will find the reference), and this notably excludes some rather close human relatives, such as the great apes.
Neurology  Language is a more complex phenomenon than speaking. The most notable difference between the animal and the human communication is that the former typically uses a set of signals with hard-coded meanings, whereas the humans can construct an infinite number of phrases coveying complex meaning. The study of language from this angle has been traditionally carried out by linguists and computer scientists rather than biologists, under the general name of universal grammar (but may also appear under such terms as transformational grammar, generative grammar, etc. ) However, these studies have a direct biological implication that humans possess a genetically coded set of rules for sentence/phrase structure, which is independent on their native language. This clearly requires certain degree of neural development, which the language-less animals do not necessarily possess.
Psychology  Finally, language requires significant psychological development - even in humans. Feral children often fail to acquire language competences. Similarly, the extensive studies on small chidlren in care homes (starting from the works by Anna Freud and Melanie Klein from more than a hundred years ago) demoinstrated that children deprieved of normal human contact have significantly diminished speaking abilities. Getting back to the OP - one can therefore question whether an animal could concievably be exposed to normal human environment to facilitae development of linguistic ability similar to that of a human.
See here and here for more background.