Ten of the 13 provincial governors are grandsons, all with administrative experience, some with genuine talent and almost all sons of kings.
Similarly, there are grandsons holding prominent positions in some key Saudi ministries.
A short list of third-generation princes who could be king includes Khalid al-Faisal, governor of Mecca and son of the respected late King Faisal; Muhammad bin Fahd, governor of the oil-producing Eastern Province and son of the late King Fahd; Khalid bin Sultan, deputy minister of defense and son of the late Crown Prince Sultan; and Muhammad bin Nayef, deputy minister of interior for security and son of the late Crown Prince Nayef.
How would a new-generation monarch be selected?
Recognizing how large and divided the royal family had become, in 2006 King Abdullah established an Allegiance Council comprising each of his remaining brothers or in the case of deceased brothers, each one's eldest son.
This council of 35 princes is intended to represent the entire Saud family in the selection of a crown prince to succeed the one who automatically ascends to the throne upon Abdullah's death. Each member of the council would have one vote; in a country that has no democracy, it would at least be a form of family democracy.
Abdullah, who exempted selection of his own successor from this process, already is on his third crown prince, each of whom he personally chose and two of whom died. As a result, the council has met only once: at its formation, when it swore fealty to king and country.
Many Saudis fear that the Allegiance Council process will die with King Abdullah - and with it the hope of a smooth generational succession.
Family feuds are not an idle worry. The Sauds have ruled Arabia on and off for more than 250 years. Infighting among several brothers ended their rule in 1891 and forced into exile a teenage Abdul Aziz, who later returned and founded the current kingdom.
On his deathbed in 1953, the long-reigning Abdul Aziz forced his two eldest sons, Saud and Faisal, to swear to avoid a repetition of this history.
The admonition fell on deaf ears. The two brothers quickly began quarreling, and their feud continued for more than a decade before Faisal, with the backing of family and religious leaders, forced his elder brother into exile.
Aware of this history, Saudis can only watch and wait, exerting no influence on succession decisions but aware that rivalries could break out and a royal house divided might not stand.
Saudi society now bears little resemblance to the passive populace of even a decade ago. Thanks to the Internet, Saudis know about life inside their kingdom and in the wider world, and they resent the disparities they see.
Fully 60 percent of Saudis are under 20 years old. They know that 40 percent of Saudis live in poverty; 70 percent can't afford to own a home; and 90 percent of workers in the private sector are foreigners, even while unemployment among 20- to 24-year-olds is nearly 40 percent.
Saudi men won't take the lower-skilled jobs for which they are qualified, and well-educated Saudi women are not allowed to take jobs for which they are qualified.
Most ordinary Saudis aren't demanding democracy, but merely a more efficient government and a more equitable distribution of the oil riches that they believe belong to the country, not just to the royal family.
It is far from certain that a ruler from the new generation could meet these demands, however modest they may seem. What is more certain is that the diminishing line of elderly brothers cannot.
So for the foreseeable future, the royal Saudi 747, richly appointed but mechanically flawed, flies on, its cockpit crowded with geriatric pilots. The plane is losing altitude and running out of fuel.
On board, first class is crowded with princely passengers, while frustrated Saudi citizens sit crammed in economy. Among them are Islamic fundamentalists who want to turn the plane around, as well as terrorists who aim to hijack it to a destination unknown.
Somewhere on board there may be a competent new flight team that could land the plane safely, but the prospects of a capable pilot getting a chance at the controls seems slim.
And so the 747 remains in the sky, perhaps to be hijacked or ultimately to crash.
Karen Elliott House, a Pulitzer Prize winner for coverage of the Middle East, is the author of the forthcoming "On Saudi Arabia: Its People, Past, Religion, Fault Lines - and Future," to be published next week. This column first appeared in The Washington Post.