Powerful and well-respected leadership, while the X country is having a lot of political dissent and divisive issues other than with the rebels, that distract the populace.

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If the project's to thrive, most important is a strong sense of distinct shared identity and a competent political leadership capable of uniting the country and winning international recognition. Anyone can declare independence and set up a government: that's the easy part; making it last is another matter, and the conditions for success and for effective statehood can't be conjured out of nothing.

For starters, they would need their current government to agree with that plan.

Or they would have to declare independence while their government is actively engaging in genocidal policies against them.
In that case, this fact needs to be widely recognized. They might even receive international help for their struggle.

Another option is declaring independence when the previous country is already falling apart.

In any case, for a successful declaration of independence, the group seeking independence would need to be a distinct ethnic, religious, or linguistic community that lives in a specific region and is a majority in that region.