No. Rabbits and chinchillas can't have offspring together. "Why not?" you may ask. Well, it all boils down to genetics and evolution. Although the rabbit and the chinchilla do look very similar, they are not really very close relatives. They are both mammals, and so had a common mammalian ancestor millions of years ago, but they are classified into two distinct groups today. The chinchilla is a rodent, in the the same large group as mice, rats, gophers, guinea pigs, etc. The rabbit, on the other hand, is what's known as a lagomorph, and belongs to a small group which also includes the hare and the pika.
Millions of years of evolution have resulted in the rabbit and the chinchilla having a very different genetic makeup, and since genes are so important to the reproductive process, this makes a mating between the two species impossible. In order for a mating to be successful, the male sex cell, or gamete, called a sperm, must fertilize the female gamete (egg). In mammals, as well as many other animals, numerous safeguards exist to prevent eggs from being accidentally fertilized by the sperm of a different species, so it's doubtful that the rabbit sperm would be able to enter the chinchilla egg, or vice-versa. Moreover, in a normal mating, the DNA brought in by the sperm combines with the DNA already in the egg, and serves as the genetic blueprint for making a new organism. For example, humans inherit 23 chromosomes from each parent, giving them a combined total of 46 (23 pairs). Different species have different numbers of chromosomes, but as you might imagine, it's very important to inherit the same number of chromosomes from both parents. Rabbits have a total of 44 chromosomes (22 pairs), while chinchillas have a whopping 64 chromosomes (32 pairs). All of that extra, unpaired DNA could cause big problems if a rabbit gamete and a chinchilla gamete ever managed to somehow get together. The cells of this new "rabchilla" organism would not be able to divide properly, and embryonic development would arrest very quickly.
