In my world the empire uses a strict racial caste system that places criminals, poor people and the minority race, the Sublorans, in the lowest class. This has been going on for a few dozen decades, and you know what? The lower caste is sick of it! They created a rebellion called the Black Mane that plans on killing and overpowering the empire, while another, smaller group of the lower caste, The Roost, just wants equality for the two races. Could these two rebellions for the same cause exist? If not, what changes do I need to make in order to achieve it?
Of course it can. All it takes is groups of people with different levels of extremism.
The way I see it playing out is the rebellion starts off with The Roost being the main rebellion. A few people start asking if they're really doing enough - "They've held us down for so long, what makes anyone think that they'll just let us be equal? The only option is for US to be in power, to remove them entirely!" Over a bit of time this viewpoint permeates through the rebellion, feeding off of the generations of inequality, until they finally split off as the Black Mane when they ultimately decide that their two goals are incompatible.
One look at Syria should make this question redundant
I've lost count of how many armed groups there currently are, most of them are at war with each other as well as the government. Some of them have the same underlying aims, they merely want to be the ones in charge at the end. At least one is just crazy. A couple of armed groups are simply fighting for their own survival, the list goes on. The West is backing at least two, while being openly at war with one. Russia is backing the government. Turkey seems to shoot at whoever they dislike most this week.
You're really not limited to only one rebellion, far from it.
Looking at a great source of inspiration - Monty Python's Life of Brian - we find that co-existing rebellions for the same cause are the most natural thing.
The Judean People's Front, the Judean Popular People's Front and the People's Front of Judea essentially fight for the same thing, nothing less than have the Romans dismantle the entire apparatus of the Roman Imperialist State within two days. However they seem to be not able to put their efforts together:
BRIAN: Brothers! Brothers! We should be struggling together!
FRANCIS: We are! Ohh.
BRIAN: We mustn't fight each other! Surely we should be united against the common enemy!
EVERYONE: The Judean People's Front?!
BRIAN: No, no! The Romans!
Certainly! In fact, there's real life examples today constantly forming. Consider ISIS and Al Quaeda. Their agendas similar enough that many don't even know the differences, and yet ISIS has supposedly shunned Al Quaeda for being too aggressive in their approach to the West. Or for that matter, just look at the history that created ISIS and try to figure out which rebel group is on which side!
You could look to Ireland for example similar to what you are describing, gaining its independence from Britain. Although the split happened later in the rebellion than yours appears to have, it could be still used for inspiration.
There were divided opinions on whether to accept Britain's terms of independence (the Pro-Treaty "Free State" group vs Anti-Treaty republican group). Although a vote ratified the treaty, the anti-treaty group were strong enough that there was a split in the cause which would go on to start a civil war after the Free State was gained, and be an emotional topic for decades, and resonates even now (nearly a century later) in the North.
It would be quite easy to translate this to your situation, perhaps The Roost began to make progress but were being forced to accept certain terms that The Black Mane would not accept.
The purpose of rebellions differs depending on where you sit. From the perspective of the supporters it is the idea both of identity and who they want to follow next. From the perspective of rebel leaders it is a lot less to do about whatever the rhetorical cause may be and more about creating a new position of power for themselves.
Two rebel leaders have the same idea, in different parts of the country, and start their insurgency. Once they become aware of one another they will work together -- but only so long as the existing government is the overwhelming threat. The moment the government is genuinely threatened by the ongoing rebellion (which was up to now a fully joint operation) then two rebel groups will factionalize and turn on one another -- it is impossible for either leader to want to make the outcome of all that they have risked a prize to be captured by the other.
This is why civil wars and dictatorial deposition often winds up creating two countries, not one.
An alternative outcome (and one that is not uncommon) is likely when both leaders realize that though neither wants the other to come into power the geography of the country mandates that it be maintained as a single political body lest the result of a fracture be rendered vulnerable either to continued rebellion (from new fronts, for new reasons) or indefensible to external aggression. In this case it is common for a sort of power exchange agreement to come into play (regardless what the public rhetoric may be).
An example of the first form is Libya. It is only a "single country" now because the West doesn't want to look too closely at it. It is currently (at least) two distinct nations -- the actual number of seats of power depends on who you are asking and what time of day it is. The counterexample is Nigeria -- where there are technically "democratic elections" but those in power are not so foolish as to mistake the holding of elections for actual democracy; in Nigeria the northern tribes have a deal with the southern "rivers" tribes that each presidency and vice-presidency will alternate between the two.
Somewhere within your world you would probably need to explain the history of why the caste system was put in place. The reason could be economical (job scarcity, resource limitations, etc.), social (crime, overpopulation, etc), political (elites vs native populations, politicians vs workers, etc.) or geological (being on different continents, living underground, settlements far from tech development, etc) Two revolutions can actually have multiple factions who might be fighting among each other in order to attain maximum benefits for themselves. These differences might be physical, ideological or based on methodology on how they express their displeasure (via violence, lobbying governments or Ministries, etc.) Also keep in mind whatever one group does, automatically effects the entire system, so while one group might go for peaceful co-existence, another more militant group might force them into a war based on attacks on soft targets, which leads to negative consequences for the entire caste, whether everyone participated or not. Like many others have quoted, studying real-life examples is probably the best way of predicting what is necessary for such a fictional recreation, based on what could happen versus what you believe needs to happen within your story. Don't forget the impact that individuals can have in groups where violence occurs (I am referring to real life individuals who fought for and promoted equality like Gandhi, President Lincoln, Mother Theresa) vs revolutionaries (Malcolm X, Che Guevara) - PS: I am only quoting singular examples; many more exist). Your characters' motivations can be good or bad, depending on their experiences, as long as it makes sense in the broader bigger picture of the world/universe you are creating. Studying wars, revolutions and uprisings will most likely give you the most information regarding real-life consequences or possible consequences of those events.
An excellent counter example is South Africa
I say "counter example" because the "revolution" resulted in a mostly peaceful transition to a legally (if not economically) democratic society, implying multi-faction revolutions can not only exist, but also be successful.
There were many different struggle factions, split along various lines (race, class, ideology, aggressiveness). They bickered and fought amongst each other, but ultimately worked together, although it took a monumental level of coordination and leadership for this to happen. Partially, this was made easier because of the level of violence and repression, but in the 80s and early 90s there were large violent battles between Inkatha and ANC supporters, where Inkatha was allegedly being funded by the apartheid government.
Also keep in mind that revolutions and wars have knock-on effects for societies - deep seated hate and prejudice against the aggressor leading to hate crimes against innocent parties, possible famine for farmland destroyed, which leads to desperation and possible barbarism further down the line, administrative and legal reforms, civil rights promoted or downgraded depending on societal issues or pressures, technological regression and public infrastructure crumbling as key infrastructure is destroyed, riots and looting as central government falls apart, reprisal killings and secret tribunals in extreme cases of a military who has gone over the edge and become a force unto themselves, etc. There is plenty of literature available online based on culture during conflicts and post-war societies. (WW1, WW2, Vietnam, Korean War, American Civil War, Latin America takeovers and coups d'État), African colonial wars, ancient Greek and Roman wars of conquest, etc.)
It might be weird to compare your idea with Feminism but actually it fits pretty well. The majority of feminism (and also the real concept of it) is for equality of all Gender, but there is also a (albeit much smaller)extremist group that also call themselves feminists even tho instead of fighting for equality they fight for and believe in the superiority of woman. Not really a rebellion with physical fighting but it shows that this concept is not new and you can definitely use it because it is very realistic.
The first item on the agenda of any Republican committee is the split — Brendan Behan.
There are several variants of that quote attributed to him, but it's certainly true, and far from limited to the Irish Republican movement.
Really, it would be the opposite that would be unrealistic. If you're going to have your rebellion as a cohesive whole, then you're going to need some sort of explanation as to why there aren't any splits.
I suggest posters:
(Source: Hark A Vagrant!)
More seriously, the most realistic way in which a rebellion could be cohesive is the way the enemies of [insert your country here] are cohesive; they're actually full of splits, but the politicians and media of [insert your country here] portray them, and even some groups who don't care either way about [insert your country here] as if they are a unified whole, and hence that's the impression people often have. This can be a self-fulfilling policy, because it tends to lead to the sort of actions that puts fighting against [insert your country here] higher up the agenda. Like how the supporters of the two gentlemen in the cartoon above found more common ground after they were both executed. Or how the Viet Cong stopped citing the leaders of the US War of Independence as heroes, and so on.