As a species evolves into a new species (say that generation X gives birth to gen. Y, and gen. Y gives birth to gen. Y+1, etc.), is it generally accepted that gen. Y+n will gradually become less capable of reproducing with the species of gen. X, as n increases?  In other words, there's no precise moment when a generation all of a sudden becomes incapable of breeding with its ancestor's species.  It's a very gradual process taking millions of years, correct?


Yes that is the general idea. In fact there can be no single mutation in a new indivudal which cuts it off from the old species and starts a new one, because were that to happen it would mean that individual and that individual alone would be different, with no-one else to reproduce with!
Generally speaking, while there are many different methods by which speciation can occur, it is often the case that two identical populations (X) are separated first, once they can no longer trade genes with each other the populations can slowly start to diverge one, but try and remember that BOTH populations will be diverging from each other and their ancestral form, try not to think of Y being further away from X, but both populations moving away from X, X ->Y, and X-> Z.

As for the speed as to which this occurs, this is down to many different factors, the population sizes (small populations will change faster as new muatations can spread throught the entire population quicker, this means that for instance if a few individuals from the mainland colonise an island they will move from X ->Y very quickly, while the mainland population will often take longer to change), the generation times, any selection factors etc. Which means that yes generally this is a process which takes millions of years, but under the right conditions it can take a few thousand years rather than millions. It still seems like a very long time, but in terms of evolutionary history, this is rapid change!

well, first of all, it need not take "millions of years". Usually it takes hundreds to thousands of generations. In organisms where a generation takes a few hours (bacteria), you can get speciation rather quickly.

Another factor, as pointed out by Phil, is population size. Here, odd occurences such as animal becoming geogrpahically isolated, e.g., by drifting to islands, are typical causes of "speeded up" evolution: very few individuals create a "new" population that has initially very little competition, grows rapidly but overall remains small (a few hundred or thousand individuals for birds, e.g.), and creates enormous intraspecific competition.

Also, it is theoretically possible even in highly complex species that the inability to breed with the parent generation happens in a single mutation. If that happens twice, in one male and one female, within a sufficiently short time frame that the two can breed togehter and have fertile offsprings, you can get speciation as a single event, which is not gradual at all (although the proportional differenc ebetween the two species is minuscle). A theoretical example is a specied of songbird that sing only in the morning, and finds mates through singing. If one individual, through a mutation, sings in the evening instead, and finds a mate receptive to evening singing through the same mutation, you effective have two distinct reproductive communities = species. But that's very improbable. Given the large number of species out there it is bound to occasinally happen, though.