Actually, yes. Broadly speaking, there were two types of Jews living in Eretz-Israel in the nineteenth century before the first Zionists arrived: small Ashkenazi Orthodox kollelim (communities supported by donations from their communities of origin abroad), and Sephardi Jews who were the descendants of Spanish Jews who had reached the Holy Land, directly or indirectly, after the expulsion of the Jews from Spain and Portugal in the 1490s.
The latter group were the overwhelming majority. They spoke Ladino (Jewish dialect of 15th-century Spanish) and Arabic, were well integrated in the local population and mostly prosperous, and the wealthy among them also had their children learn French at school.
Today they are known as Sfaradim Tehorim (“Pure Spanish [Jews]”)—or the Hebrew initials ס”ט (Samekh-Tet). Because they had lived in the country (mostly Jerusalem) for centuries, they were, and to a large extent, still considered a kind of aristocracy.
Israel’s first Sephardi President, Yitzhak Navon, was a scion of one of those families, and when, in my twenties, I went out with a girl from the Castel family (a well-known Samekh-Tet clan), my mother was hugely pleased, and said, “Well—you could do worse!” (Alas, I did not marry her, in the end, but another Sephardi girl whose family spent the last 500 years in Turkey—almost as good…)
But by far the most telling answer to your question is the common practice of stating how many generations one’s family has lived in the country (as you can tell from my own measly “Third generation” self-designation in my Quora.com profile). Israelis of six generations and above carry their pedigree with the same nonchalant pride that British aristocrats allude to their titles.