I think the definition of a species is that its members can produce viable, fertile offspring. So in theory, the possibililty of them reproducing successfully would be unlikely, although certain combinations of 2 species have been known to produce hybrid, non-fertile (ie. dead-end) offspring (eg. mules).
In the case of domestic dogs, they are all breeds of the same species and can therefore breed to produce viable offspring. I think the situation is that despite distinct characteristics having arisen between breeds by extensive artificial selection, chromosomal number and sufficient genetic similarity are conserved within members of the species. That allows cross-breeding to occur successfully, provided any physical difficulties can be overcome!