Sort of. It depends on what you mean. For the sake of simplicity I'll confine my discussion to the Sunni religious tradition when I mention the ulama, though to complicate matters, "Sunnism" did not exist for much of the period I'm discussing "orthodox Islam" might be more accurate.

Though it is debated at the margins, Sunni religious thought has adopted the notion that even a corrupt ruler is to be obeyed. This is closely related to the concepts of fasad (corruption, discord), and fitna (civil strife, dissension). It's a concept that applies equally to a caliph or a sultan.

The Quran makes clear that those who sow political discord are to be punished by death. Sura al-Maidah, 5:33:

The only reward of those who make war upon Allah and His messenger and strive after corruption [fasad] in the land will be that they will be killed or crucified, or have their hands and feet on alternate sides cut off, or will be expelled out of the land. Such will be their degradation in the world, and in the Hereafter theirs will be an awful doom;

That being said of course, there have been such moments of civil strife, referred to as the First, Second, Third, Fourth and Fifth Fitnas.

Brief overview:

The First Fitna: War of succession after the assassination of Uthman, the third caliph. Ali is declared caliph in Medina. Uthman's cousin Muawiyah, governor of Syria, rebels. In a negotiated treaty, Ali recognizes Muawiyah but is assassinated by his own men. This establishes the Ummayad Caliphate, and is arguably the most important.

Second Fitna: After the death of Muawiyah, ibn al-Zubayr revolts in Mecca. He's eventually beheaded and crucifed under Abd al-Malik, but is a marginally successful anti-Caliph (if you will) for about 12 years.

Third Fitna: The Abbasid revolution. The Ummayads are overthrown and their remnants flee to Spain and a couple centuries later claim to be a caliphate based in Cordoba.

Fourth Fitna: Basically, Harun al-Rashid bungles his own succession, and creates a pseudo-division of the empire between his two sons. One of those sons, al-Mamun, revolts and kills his brother al-Amin.

Fifth Fitna: The Anarchy at Samarra: Basically another bungled succession after the murder of al-Mutawakkil, but this time with elements of a Turkic slave soldier junta as well.

Each of these Fitnas involved competing claims to the caliphate, including sometimes the creation of counter-caliphates. There are of course other examples. The 10th century saw Ummayad Caliphate of Cordoba, the Fatimid Caliphate in Egypt, and the orthodox Abbasid Caliphate in Baghdad.

By implication, the acceptance of the ulama in each country for this state of affairs was a rejection of the other caliphates. This cut across sectarian lines for both the religious and military institutions of Islamdom. So again in the 10th century you end up with the bizarre situation of nominal Sunnis reciting the names of the Shia Fatimid Caliphs at Friday prayer, including in the holy cities of Mecca and Medina where local (again, nominally Sunni) leaders voluntarily recognized Fatimid rule. Likewise, the Shia Buyids who ruled much of Greater Persia in the 10th century preserved the Caliphal authority of the Abbasids.

There were also examples of the ulema as a whole broadly taking sides. In the Abbasid revolution against the Ummayads, the religious establishment broadly backed the Abbasids.

What I'm not really aware of aside from out-and-out heresies is an example of the rejection of the caliphate by the ulema being the driving factor behind some sort of political or military revolt against the caliph, except, as in the case of the sharifs of Mecca or certain tribal revolts, where elements of religious authority and scholarship were simultaneously vested in people with tribal or other political power.

I think that has a lot to do with development of both the Caliphate and ulema both conceptually and as institutions. Neither arose "organically" at the time Islam emerged. The ulema developed out of basic legal disputes about what Islam was. The founder of the first "school" of Sunni legal scholarship, Malik ibn Anas, wasn't born until almost a century after the death of Muhammad. The major hadith collections weren't compiled until the 9th century, which is around the same time that the notion that the primary sources of law were to be the Quran and the hadith after Shafii.

What the Caliphate meant and what its power and role consisted of likewise shifted over time. Even the word itself was in dispute. "Caliph" means successor or deputy, but deputy of what? Are they "Khalifat Rasul Allah" the Successor of God's Messenger [the Prophet Muhammad]? Or are they "Khalifat Allah", God's own deputy on earth?

One of the most important confrontations between the ulema and the caliphate was over precisely such questions of their respective authority and role. After the Fourth Fitna, the Caliph Mamun in the 9th century attempted to enforce a doctrine that the Quran was "created" by God at some time and then revealed to Muhammad, in opposition to the idea that the Qur'an was eternal and uncreated despite its revelation to Muhammad at a specific point in time.

In order to enforce his position, Mamun instituted the "mihna", a religious inquisition that required all of the judges of the empire to publicly declare their support for the doctrine. This was rejected by the traditionalist (Athari) portions of the ulema, most notably Ahmad ibn Hanbal, founder of the eponymous Hanbali school of jurisprudence, one of the four Sunni schools of Islamic law.

Though carried on to an extent by his successors, the Mihna was ended and the doctrinal insistence on the createdness of Quran was eventually withdrawn by Mutawakkil and the mihna abandoned.

For what it's worth, ibn Hanbal's own position on this question was that:

...The Friday worship, the (celebration of the) two feasts, and the Pilgrimage (are observed) with the authorities (sultan), even if they are not upright, just and pious. Taxes (sadaqat, kharaj, etc.) are paid to the commanders (umara), whether they deal justly or wickedly.... Those to whom God has entrusted your affairs are to be followed...and not opposed by your sword....To keep aloof (from both sides) in civil war (fitna) is an old custom whose observance is obligatory. [—Watt, Formative Period of Islamic Thought]

As I've mentioned, this is basically the normative position of Sunni scholars throughout the history of Islam.

The mihna is noteworthy though because its failure marks a kind of final triumph of the ulama. The independence of the ulama from state authority at the same time that the state relied on the ulama to determine the law is a critical feature of the Abbasid caliphate. Where the Ummayads, on the basis of a claim to divine authority, acted and legislated in a way that we might call secular, for the Abbasids the ulama basically get to say what the law is, even if the Caliph still gets to appoint judges. The mihna demonstrates just how circumscribed the caliph's authority was, even as the prior century was a period of Caliphal absolutism building on the Ummayad model. That the succeeding century saw the Abbasid Caliphs become figureheads enthralled to secular military commanders I don't think is a coincidence.

That division of responsibility wherein the ulama are basically responsible for society while the military elites and the Sultan collect taxes, maintain security, fight wars, and govern the state, pertains until the Mongol invasions, the sack of Baghdad and consequent end of the Abbasid caliphate of Baghdad in 1258.

While the Mamluks give shelter to a member of the Abbasid family whom they claim is Caliph and have his name recited in the mosques, that claim is basically rejected by the wider Islamic religious community and the Abbasid caliphate of Cairo faded into obscurity until it was ended conclusively by the Ottomans after their conquest of Egypt in 1517.

This period in which the majority of what was now more coherently understood to be a Sunni community did not have any recognized Caliph resulted in quite a lot of retrospective consideration for what the Caliphate was.

Thus two thinkers as different from one another as Ibn Taymiyya and Ibn Khaldun agree in declaring that the caliphate ceased after the rashidun and that the sovereignty exercised by the Ummayads and the Abbasids had never been more than a sort of "royalty." This was their way of expressing how they were struck by the contrast between the theory of the caliphate as it had been formulated by al-Mawardi and the reality that they saw. It may be added that the lawyers of the Hanafi school were also of the opinion that the true caliphate had lasted no more than 30 years, up to the death of Ali. [—Encyclopaedia of Islam]

So freed from caliphal authority for less than a century, broad swathes of the ulema basically rejected every caliph after Ali.

Running out of characters so TBC.