Anabela,

You've asked an interesting question but one with many possible answers.

First, let's look at Neanderthals. From Wikipedia:"A 2013 study of Neanderthal skulls suggests that their eyesight may have been better than that of modern humans, owing to larger eye sockets and larger areas of the brain devoted to vision. Neanderthals are known for their large cranial capacity, which at 1600 cm3 is larger on average than that of modern humans. One study has found that Neanderthal brains were more asymmetric than other hominid brains. In 2008, a group of scientists produced a study using three-dimensional computer-assisted reconstructions of Neanderthal infants based on fossils found in Russia and Syria. It indicated that Neanderthal and modern human brains were the same size at birth, but that by adulthood, the Neanderthal brain was larger than the modern human brain".

Here we have a hominid that (a) cross-bred with modern humans; (b) had bigger brains than modern humans, and (c) had more brain area devoted to one of the most-developed human senses, vision. Are we sure that they had more limited mental capacities than modern humans? Some may argue this based on less rich "cultural" remains in the archeological record. But, by this criterion modern H. sapiens must be much more intelligent than our 30,000 year old ancestors, which is unlikely. 

My second answer relates to the so-called "theory of mind". It has been proposed that social animals develop brain structures allowing them to better predict how other members of their social group will behave. If you do a search for "mirror neurons" you will find lots of interesting studies on this subject, with comparisons of humans to other primates and other social non-primates.

My third answer is to suggest you look at some of the parts of rationality that seem uniquely human and how we are beginning to understand their basis in the brain. One good example is numbers. There have been fascinating articles published in the last few years looking at how the human brain deals with numbers, and there have been discoveries in much simpler species (chickens, for example) showing they have  some parallel capabilities.

My fourth answer perhaps comes closest to the spirit of your question. The "ontogeny recapitulates phylogeny" approach has some parallels in the work of Piaget and his successors on the growth with age of children's mental capabilities. A look at a recent article on Piagetian stages and their non-human analogues might help you.