Do human-ape hybrids exist?
Although this question sounds easy to answer in the first place, it is not. The main problem between humans and the apes we are most closely related to is that we have 46 chromosomes, while the apes all have 48. At some point in the evolution of the humans the ancestral chromosomes 2 and 4 fused together to form our chromosome 2, leaving us with only 46 chromosomes (see reference 1).

This makes offsprings much more complicated, but not completely impossible. A good example for this are crosses between donkey and horses (donkeys have 64 chromosomes, horses 62) to get mules (male donkey crossed with a female horse) or hinnys (male horse crossed with a female donkey) which subsequently have 63 chromosomes and are infertile. There are even crosses possible between horses and zebras which even differ wider. Zebras have between 32 and 46 chromosomes (depending on the species) but still zebroids are possible. See reference 2 for some more details.

So far there has been no successfull attempt to produce such hybrids, although there have been some attempts to make them. Some information can be found in reference 3.

Besides this problems (which are also due to ethical restrictions), there is a study available that at least humans and chimpanzees interbred for at least 1.2 million years after diverging and still produced fertile off-springs. See references 4 (for a summary) and 5 (original paper) for details.

Based on this evidence, I would say it is possible for humans and apes to produces off-springs, however, this is relatively unlikely and they will probably be infertile.

References:


Origin of human chromosome 2: an ancestral telomere-telomere fusion.
The chromosomes of two horse x zebra hybrids; E. cahallus x E.
grevyi and E. hurchelli
Blasts from the past: The Soviet ape-man scandal
Human Ancestors May Have Interbred With Chimpanzees
Genetic evidence for complex speciation of humans and chimpanzees