Is it possible for one to be Anglican and Roman Catholic at the same time?
Well, yes and no . . . depending on where you're standing.
The Anglicans (at least the "high" ones) have always maintained that they have just as valid Apostolic Succession as the Roman Rite Catholics. The Catholic Church disagrees . . . hence the Bull Apostolicae Curae.
Reader's Digest Condensed Version: when Edward VI was king, his radical Protestant advisers changed the words of the consecration of bishops (removing references to the sacraments and the Holy Ghost and also the intention to supernaturally ordain) and by 1662 when they changed them back, a century had passed and the succession was extinct.
Canterbury and York fired back with the response Saepius Officio, which basically is "Is Not Either!" Their position was that Anglican Orders, having originally stemmed from the Catholic Church, remained valid despite the change in wording.
And there the matter stood for almost a hundred years, with the Anglicans maintaining they were Catholic (if not Roman) and the Catholic Church saying, 'uh uh'. Some Anglicans maintain that their Apostolic succession actually stems from the ancient Celtic Church, but that's just wishful thinking, because there are no records of ordination.
There was talk in the 60s and 70s of revisiting Apostolicae Curae, presumably with the idea of coming up with some sort of conditional ordination . . . but the antics regarding ordination of women and homosexuals, etc. put the quietus to that.
200 or so years of family tradition is a hard thing to shake.