Would it be possible to breed chimpanzees as intelligent as humans if one mated only the smartest chimps from each generation for many years?

Yes, definitely, but it would probably take a long time. Also, you'll have to define "intelligence" very carefully.

This is not possible for multiple reasons, two of which are highlighted here. First, human intelligence is a product of a mosaic of behavioral and cognitive traits that evolved in the context of a tightly wound feedback system between biology and culture. So, the question of breeding smart chimps to reach human intelligence can only make sense if "intelligence" can be imagined as a product of a single (or a few single) traits, and also discounting the interplay between biology and environment (and in the later stages of evolution, culture). This certainly is not the case. Second, there is no reason to suggest that these traits in chimps which we recognize as denoting "intelligence" have any relationship to the mosaic of evolutionary traits which we humans, over time, accumulated, some of which may have erupted out of random mutations. In this way, even if you define intelligence in chimps and identify the aspects of the genotype which correspond, it is totally unlikely that the favoring of these traits will result in a human-like intelligence. Hope this helps.

There are assumed to be many genes favourable to high IQ, both in humans and other species.   By breeding, you could maybe produce a breed that had all of them in all individuals of that breed.  But you could also find that some of the high-IQ genes have nasty side-effects, as happens with selective breeding of dogs.   This would take centuries, and you would still just have a smarter chimp.  Humans have accumulated a lot more relevant genes in the millions of years since we split from chimps.   Long before that, we are likley to understand the genes involved.  Then we could do it much quicker using genetic engineering.  (Assuming we want to, of course.)

It took millions of years for us to evolve. If by "many" you mean anything not at least within a few orders of magnitude, and you are not directly manipulating and changing genes (effectively creating a much higher mutation rate to speed up the process) the answer is assuredly no. To have our intelligence the same mutations as occurred in us would have to spontaneously occur again, without other negative mutations getting in the way.  This takes a LONG time by random chance.

   The short answer is yes.     The longer answer is that this would be an extremely difficult process, taking many generations probably several thousand years in which succeeding generations of selected chimpanzees would be bred to other high performing chimpanzees. Over time this would result in chimpanzees with a higher performance for the traits selected and depending upon the conservation of nearly random mutations in brain development and intelligence result in near human intelligence.