There is a simple if unsatisfactory answer to this: 'maybe'.

It might be more satisfactory to ask: 'could primates evolve to be like us in the same time it took us to make the transition?'

Evolution has no 'plan', no 'design', and therefore there is no way to tell how a species will evolve.  In particular, evolution is highly influenced by the environment - the weather, food resources, terrain and so on.  We know some primates evolved and became us, so it is conceivable that a similar set of circumstances will converge to allow another primate to become like we are today.  Convergent evolution occurs around us, such as arctic animals evolving to favour white coats and low surface areas for their masses (to reduce heat loss).

On the flipside, however, we have to remember that in the time it took us to evolve from primates to us, the other primates were also evolving.  No other primates evolved to be like us except ones that diverged from a recent common ancestor (eg Neanderthals). 

This also highlights the problem of defining 'like us' - I'd say a Neanderthal is like us, but an Australopithecus is not.  Chimps have also been evolving over the same period, and share many of the same characteristics as us.  It depends on the characteristics used to make a judgement on what is 'like us'.  It could mean social creatures with technology and language who act like us, but look nothing like us.  Take The Planet of the Apes - were the apes like us, or not?

So primates could evolve to be like us in the next 15 million years, but probably won't because there is a huge range of possible forms they could take in that time and some of these could be as successful as - or more successful than -  ours.

It doesn't look likely they will have any time to evolve. We're are pressing many of the great apes into extinction through habitat destruction, hunting and disease (they may have given us HIV but we have much more to give).

If they do survive there's really not a lot you can say on how much evolving they have to do. Species last about 10 million years on average according to the fossil record, but evolution is not yet predictable. In the absence of strong selection pressure, genetic drift means they will become genetically more diverse in small ways but not really "evolve" much. Under human-driven selection pressure, we could expect them to become more human in some ways, as cats and dogs have. We are selecting for cute, sociable and intelligent. We are unlikely to create the Planet of the Apes in any reasonable period of time, but since they can already do many of the things we used to think are unique to humans, such as use tools and communicate using sign language, they might progress from the intelligence of a very small child to a slightly more developed child in the next few centuries of living with humans.

Without humans, I can see them evolving into "our" niche under similar conditions: climate change, moving out of the forest onto the savannah and adapting to rapidly changing conditions over several million years. This is not a given, however, although it makes for good SF and fantasy stories.

Evolution started when life started and will end when life ends.  It does not finish when it arrives at some state, such as people.  If the questioner doesn't consider a chimp, for example, to already be like us, then the chances of some non-human primate evolving to be like us are vanishingly small.  If you consider a chimp to be very much like us (in contrast, both chimp and human being rather unlike a jellyfish, say) then the chances of it happening are certainty because it already did.