From personal experience and observation of various other people, I think a "rebellious" streak is necessary in order to make a clean break from childhood (where you are the recipient of lots of forces acting upon you, including parents, family, peers, and culture) and enter adulthood, where you can now make autonomous, self-directed decisions.
I think the severity of the 'rebellion' depends on the personality involved, how accommodating they were of the system beforehand, and the size of the disparity between internal inclination and external pressures, among other things. Those who are rebellious from an early age and never fully accept the environmental imprint usually have to learn the opposite sort of lesson -- how bend and accommodate others -- in the process of becoming integrated into society, while the obedient or accommodating will at some point have to make a clean break from being a child and dependent on the system before coming back to engage it as a truly independent adult. Otherwise one lives a sort of "half-life," beholden on the approval of some range of authority figures.
Usually it takes a crisis (or series of crises) of some sort in order for one to make the necessary changes toward either rebellion or accommodating/connection. The behavior pattern until that point is fairly stabilized, especially if the person has reached adulthood in the legal/physical sense, and has to be upset/challenged by something fairly large and disruptive, where lines finally get drawn and commitments to self or others get made.
This is typically also part of attachment theory, with children. It's common for the child to quickly becoming attached to Primary Caregiver (usually the mom in western culture), so after 6-12 months the child will fixate on PC and be more wary of strangers, and when the child begins to range out (after learning to crawl and walk), the PC provides an anchor/reassurance point the child keeps coming back to. Eventually the child will hopefully stretch and explore and that tether becomes longer and longer. Typical parenting involves moving from a more authoritarian mindset when the child is young to one where the parent is letting the child be more an more autonomous as the child approaches adulthood. The rebellious streak will be in proportion to the amount of force exerted by the parent to keep the child under control, so parents who equip their children to think autonomously and encourage expression of ideas and exploration rather than stifling the child will hopefully lessen the severity of the rebellious streak. But this is typically why the teenage years -- where the child has more capacity to make decisions, more awareness of decisions to be made and social influence being imposed upon those decisions, and is physically, emotionally, and mentally coming closer and closer to living independently -- are the rebellious stage. The child has to "leave the nest" before he or she can fly back, so to speak, and parents have to relinquish those bonds.
For me, I had a fractured life for much of my adulthood. I never had an actual rebellious phase in my teenage years and instead maintained the goodwill of authority figures in my life, partly to keep my life calm and partly to win approval. I felt too voiceless within my family and subculture to be able to express an opinion (all it caused was sterss and trouble inflicted on me), so typically internally I held one set of values, externally I knew what the other set of values was, and I tried to somehow express only the intersection of the two (aside from trying to flex and twist those demarcations in order to expand that intersection set -- i.e., change the environment if I could without pushing hard enough to break it). I also, like a child, did not want to lose people's approval, which I had received all my life. When I was alone or in other environments, I was able to express myself far more freely, and there was a growing dichotomy between the inner and outer even if the intersection of the two that I expressed was still part of who I was.
I finally got to a point where I couldn't handle the stress of that dichotomy, and my environment (family and social network) responded pretty much as I had expected, and I lost a lot of support. It was devastating to go from being respected by everyone and considered a font of advice and an example to, literally overnight, the black sheep who people either shun or else "patiently await my return" to their values.
For example, a lot of my family sees my stopping attending church as some sort of adolescent rebellious phase I'll outgrow, whereas actually it was far more a true reflection of my idea that spiritual values for me are not something that typically grows within institutionalized religion; I need far more freedom than that, and I hated the way that open discourse had always been stifled in my family and in church. It was also one of those situations where I wasn't yet quite strong enough to push back against the array of social forces putting pressure on me to conform, so I did the next best thing: I left. I might one day become a consistent member of some "group expression of faith" but it will be on my own terms, and a group of my own choosing rather than one meant to accommodate my family.
In that sense, then, it's very much that I had to "break away" to get rid of the lingering traces of the old so that if and when I finally reclaim some of that, it'll be on my terms and a reflection of my own choices and values, not anyone else's.