Other animals will reach human intelligence if there's sustained evolutionary pressure to that end. If smarter animals have a greater chance of reproduction, then the population will evolve to become smarter. There are some hurdles, though. About a quarter of the energy used by our bodies is consumed by our brain. Think of that - the brain is roughly 1-2% of our weight and consumes 25% of our energy. That's a HUGE penalty to overcome if food is scarce or if conservation of body heat is important. That immediately reduces the area of the planet where intelligent life is likely to evolve by a large factor.
In the history of evolution, things like vision, flight, echolocation, opposable thumbs, and many other features have evolved multiple times along distinct lineages. Human levels of intelligence, though, has only evolved once. This indicates that it probably requires some pretty specific circumstances to develop. It is also possible that having one species with high intelligence dominating the planet (humans) will in some way prevent another species from developing similar intelligence. It may be that intelligence fills a niche that once full prevents other species from being able to successfully exploit the advantages of intelligence without being overwhelmed by the disadvantage of having to support a large brain.
Of course primates are the most likely animals to evolve intelligence, but since they share a pretty recent ancestor with us that's not a very interesting case. Dolphins have a shot, perhaps, but I think that's probably fairly unlikely due to a lack of flexibility in their bodies - they are much more limited than primates in their ability to manipulate their environment (though there are some indications that they've learned to use tools to catch food without human influence) so I think their ability to take advantage of increased intelligence is limited. Some birds are very intelligent as well, but I suspect evolutionary pressure will force them as a species to maintain the ability to fly in most cases where intelligence might be selected for - so the increased brain mass would come with a great penalty that might not be something that could be overcome. Obviously there are flightless birds, but they've evolved largely in environments that offer other obstacles to the evolution of a large brain.
In any event, evolution has proven itself capable of creating some pretty surprising things, so who knows what'll happen.