For starters, Israeli is a national term and thus you can only technically be "not really Israeli" if you don't hold Israeli citizenship. I suppose one becoming a national of another nation may be seen as a type of conversion...but we don't call them converts in this case but rather patriots to that nation. Many of such people may come from Europe, America, or Africa...however, a lot come from Asia too, where Israel is. And, it's important to note that their kids born after such a move are Israeli since birth.
But I suppose you're really talking about whether or not the Jews are really the descendants of the Israelites. Israelite is used differentiate between Biblical Israel and Modern Israel, which uses Israeli. As far as we can tell, they are. The thing is that Jews since being expelled from Israel have always kept their traditions and hopes alive, and that was passed down through the generations until now, when they can return to the Promised Land in great number. This goes for Jews who traveled throughout Europe, through the Mediterranean, and even throughout the Middle East. They do NOT fake their religion, or at least not the culture and heritage that used the religion as its backbone. Thus, "fake" is really a bad term. Even if you want to mean they're not of pure Israelite descent, you have to keep in mind that the standards of the Jews (back then as well as now) do not care for such "purity" and fully allow in converts too. Still, tests have shown that even the "European" Ashkenazi Jews tend to trace descent back to the area.