The system, is not perfect. It has changed by multiple people for multiple reasons. See at the end for that. Based solely on the first film, the system in place is not designed to accommodate divergent individuals.
It only provides the illusion of choice.
The kids are raised in their factions, conditioned to believe that faction as the right one. During the test, they are found to be one of the five. But they can choose, to stay with their birth faction, go with the test faction, or the faction they want. The fourth option is to be factionless, an outcast.
As Tori explains, a series of tests (five funny enough) are given, but most end up with their birth faction:
Tori: You'll be offered a series of choices to test your aptitude for each faction until you get one result. I wouldn't sweat it. 95% get the faction of their origin.
95% of people stay in their own faction. Those that don't have rumors about them. It's a shocking thing in the faction society. And it's so unlikely that non-conformists are rare, compared to Divergents.
Of course, right before:
While it is our belief that choosing the faction indicated by your test is the best way to ensure success within the faction system, it is your right tomorrow at the choosing ceremony to choose any of the five factions, regardless of your test results. However, once the choice has been made, there will be no change permitted.
But choosing to go against the test can have consequences. Immediately after the ceremony, Beatrice and the others who chose Dauntless face multiple tests to weed out those that arn't really dauntless. Climb to reach the train. Jump off the train. Blindly jump off a building. If they don't, they are immediately factionless.
Al: What if you don't jump?
Peter: What do you think? You'll be factionless.
And then the training comes, where they weed out more. Before Tris' class, anyone who survived the training passed. During Tris class, they develop the point cut off to weed out more non-conforming. Not just divergents, but people who are really a different faction but chose Dauntless, and could bluff or bs their way in. If you don't think like the faction you choose, they will reject you, regardless of what the kids were told that they had a choice. Conform or
die be removed from civilized society and become a factionless hobo.
Now during the movie, the system was changed to hunt down divergents as well as turning Dauntless members into brainless weapons.
So the reason Four implies that Tris is Divergent, aside from his only status as Divergent and his instructor giving him the same advice for the same reason, is because everyone is on the look out for Divergents. It's more likely that someone is Divergent than it is for someone to choose outside of their faction. And they treat them the same as Divergents anyway.
The system has been corrupted to the point where they are overzealous about finding Divergents, that even people jumping factions would be treated as Divergent, even if they arn't. Like modern day Islamophobia or Racism, where people who may superficially resemble the target get affected. Think Sikhs vs Muslims.
Tori: My brother was like you. During the second stage, he got good really fast. Last day of the simulations, one of the Dauntless leaders came to watch him. Then the next morning, we found his body at the bottom of the chasm. They got rid of him.
Tris: Who did?
Tori: Dauntless leadership. You can't let them find out about you.
Tris: And what if they already know?
Tori: Then you're already dead.
In Insurgent, when Erudite creates the Divergence Tester, Eric finds someone that barely counts as divergent.
Eric: Hello. [he scans Uriah]
Scanner: Divergent ten percent.
Eric: Hm. Shit. You' re almost normal. That's boring. [he suddenly shoot Uriah in the head, killing him]
Prior to the series:
Before the five faction system was created, as we find out in Insurgent Allegiant, the system was designed to create divergents as a sort of salvation for humanity. At some point that is forgotten and the five faction system is enforced. And during Divergent, Jeanne and others actively try to stop divergence from existing. The system has been corrupted from its original intent, so what we see in Divergent is a divergence from what it should have been. It's possible that at the beginning, someone could choose against their intended faction, and leeway would be given in their initiation tests.
Caveat: We basically only know how Dauntless would handle it, and only during Four and Tris' classes, and the other factions are left mysterious. As the military branch, Dauntless is likely more aggressive in their training, and anyone not acting like a good little soldier would be removed. Candor likely finds out quickly if someone can't be truthful (hence the truth serum hurting Tris in Insurgent). Erudite picks off anyone not smart enough, while the other two can be more welcoming (the stiffs feeding the factionless while the gardeners giving the rebels refuge as long as they don't fight).