The sort whose existence I first learned of is formally Catholic but materially Protestant. Although they nominally belong to the Catholic Church, their attitude toward the teaching authority of said Church is virtually indistinguishable from that of Protestants. Thus they reserve the right to decide for themselves which teachings are "essential" expressions of the faith-once-delivered and which are revisable human opinions. That attitude effectively reduces the entire content of revelation to human opinion, i.e. what Newman called "private judgment." As I argued in my essay Faith, Private Judgment, Doubt, and Dissent, that is the real "Protestant principle."
As a staple of orthodox Catholic apologetics, CINOs (Catholics in name only) are too familiar to be worth belaboring here. But there are two other sorts of PTC whose subtlety is often missed in ecclesiological debate.
One is the sort of PTC that Richard John Neuhaus was while still a Lutheran pastor who had not yet seriously considered conversion to Catholicism. He believed, rightly, that he had to have good reason not to be a member of the Catholic Church; accordingly, he believed that his brand of Lutheranism, the "evangelical-catholic" brand, was as close as one can get to professing the faith of, and trying with his co-religionists to embody, the one, holy, catholic, and apostolic church referred to in the Nicene Creed. Eventually, of course, he came to realize that such a thing was a mere idea—a beautiful idea, an idea more or less shared by some very intelligent members of his party within Lutheranism, an idea that had summoned forth his and their loyalty. Now as recounted in his recent book Catholic Matters, he eventually realized that his loyalty was simply to his idea, "which is no loyalty at all." That's when he was set on the path to real Catholicism. But I have encountered more than one person, both in my past as a student and in my recent career as a blogger, who still manages to believe that, while they are formally Protestant, in the sense of belonging to a Protestant church, they are materially more catholic than either the Catholic or the Orthodox churches. They believe they know Scripture and Tradition quite well enough to know the faith-once-delivered in its fullness, and thus that the Catholic and Orthodox churches, unlike their own party within their Protestant denomination, have at some point fallen away from it. This is a more refined form of self-deception than you find in most CINOs, but it's just as deep. It's also important to know it when you see it, so that time isn't wasted thinking you're dealing with a garden-variety Protestant.
The third and most refined brand of PTC is what was once called a "High" Anglican and is nowadays best exemplified by "Continuum" Anglicans. Most of those are branch theorists: people, mostly men and often clergy, who believe that the one, holy, catholic, and apostolic church does not subsist in any single, visible communion but rather ramifies into the three "branches" of Catholicism, Orthodoxy, and Anglicanism. Of course the Anglicanism in question is their party within Anglicanism, which is only one; and branch theory depends crucially on the assumption that members of that party know better than either the Catholic or Orthodox churches what the ecclesiology of both churches truly entails. Like the finest vodka, this is private judgment distilled so effectively that one hardly knows when one is drinking it. It can make you drunk before you know it. That's probably why its irony is lost on those who profess it.
The antidote to all this sort of thing is the scandal of particularity. Whether you're Catholic or Orthodox, you believe that your church is "the Church." I don't know of any honest way round that. There's no way to identify the faith of the original, undivided, catholic church that does not entail identifying some visible communion as that church; otherwise, one ineluctably lands in a fantasy church woven from private judgment. That kind of self-deception will always be with us; but as long as we see it and name it, we can limit its victims and debate its perps more efficiently.