From Ask a Biologist: http://www.askabiologist.org.uk/answers/viewtopic.php?id=3250
"Although the domestic and cotton tails may attempt to breed, the embryos will usually die before birth due to the difference in chromosomes. Domestic (European) rabbits have 22 pairs of chromosomes, cottontails have 21"
So the answer is technically yes, it could result in hybrid babies, though it's very extremely unlikely any offspring will survive.
As note, there are hybrid animals with parents with a different number of chromosomes that are common, as the mule is the offspring of a horse and the donkey, for example. The horse has 64 and the donkey has 62. The mule ends up with 63 chromosomes. Due to the odd number, the mule is usually sterile, but a very few cases have been recorded of one managing to reproduce.
There also has been a recorded case of animals so distantly related that they were different genuses (one degree farther removed than species) and still bred to produce living offspring, as there were offspring produced from a mating between a babirusa and a domesticated pig when the two were put in a zoo together. http://www.macroevolution.net/babirusa-domestic-pig-hybrids.html
In conclusion, there's always a chance for an exception, but by far the most likely outcome is no viable offspring would result.
Yes as a matter of fact you can successfully breed hybrid rabbits, half domestic and half cotton tail. There's a number of breeders online that do raise them.
Our domestic rabbit has just had 4 babies from a wild rabbit, she got out into our field, we safely recaptured her and we now have babies!!! So definitely possible!