How many species can have the same scientific name?
The scientific name (better known as latin name or binomial name) of a species is unique to this species. No two species can have the same latin name. Also, a single species cannot have two different latin names. But of course, mistakes happen and we don't seem to bother too much about them (esp. when the two species are very unrelated; see What instances are there in which two species share the same binomial name? (thanks @fileunderwater and @HRA).

Of course, the above is true as long as there is agreement in the limits of the definition of a species. In absence of such general agreement, there can be some issues. Consider for example, Homo neanderthalis who is often called Homo sapiens neanderthalis. See the post How could humans have interbred with Neanderthals if we're a different species? for more information about the definition and delimitations of species.

Have a look at wikipedia > Binomial nomenclature for more information.