There are two different rationales for saying that Catholics are “not Christians.” On the one hand, you have the very conservative Protestants (extreme fundamentalists, ultra-conservative Calvinists, etc.) who would say that the doctrines of Catholicism are so completely contrary to the Gospel that they amount to a different religion. These people would vary on when this began to be the case. Many fundamentalists might cite Constantine. Modern conservative Calvinists are more likely to speak of Trent as the point where “Rome anathematized the Gospel.” Or there could be a number of points in between. All these different views account for a tiny minority of modern Protestants, though until late in the 19th century they were probably the majority.
The other, much larger, group who might make such a statement consists of evangelicals, who would usually say something like “many Catholics are not Christians, but some are.” The likelihood of a Catholic being a “Christian” would vary depending on how fundamentalist a given evangelical is. This view is based on the notion that Christianity is not at its core a “religion”–i.e., a set of beliefs and practices–but rather a “personal relationship with Jesus,” which you enter by an act of the will in which you put your trust in Jesus for salvation. At its most moderate, this is not an anti-Catholic view at all–many evangelicals would say that there are members of all churches who have not put their trust in Jesus. However, evangelicals tend to think that the Catholic Church does a rather bad job of teaching people to have a personal relationship with Jesus, so they are more likely to suspect a Catholic of “not being a real Christian,” though members of certain mainline churches such as the Episcopal Church don’t fare much better. The more conservative/fundamentalist an evangelical is, the greater this suspicion of Catholics will be, and the closer it will come to a dogmatic declaration that “Catholics are not Christians.”