Yes. Personality changes slowly and, on average, in a predictable direction: across adulthood people become more agreeable, more conscientious and more emotionally stable, and the assertive side of extraversion rises in young adulthood. Most of that change is gradual, over years, and most people keep their rank among their peers while the whole group drifts.
The evidence
The clearest evidence is a meta-analysis of 92 longitudinal studies by Roberts, Walton and Viechtbauer (2006), which tracked mean levels of the Big Five across the life course. Social dominance (the assertive part of extraversion), conscientiousness and emotional stability rose through young adulthood and middle age; social vitality (the sociable part) and openness rose in adolescence and early adulthood and then fell a little in old age; agreeableness rose most after fifty. The changes were moderate in size and continued well past the age at which personality was once assumed to be fixed.
Two kinds of change are worth separating. Mean-level change is the whole group moving, as above. Rank-order change is people swapping places: whether the most conscientious twenty-year-old is still the most conscientious forty-year-old. Rank-order stability is high over short intervals and lower over decades, so individual people do change relative to each other, but slowly. The links between traits and life outcomes that Soto (2019) replicated in a pre-registered study are consistent with traits that are stable enough to predict things and plastic enough to move.
What drives it
The evidence points to ordinary adult life: taking on work, relationships and responsibilities is associated with the rises in conscientiousness and emotional stability, and the changes are larger in people whose lives change more. Deliberate attempts to change a trait show small effects in the studies that have tried them, concentrated in the traits people most want to change.
What it means for your results
A result is a reading of now. If you take the same test in six months and the score has moved by a few points, that is usual and does not mean you have changed; the scales are not that precise. A move of a band, from the 30th percentile to the 60th, is worth noticing, and the site shows two attempts on the same strip so you can see it. If you are young, expect your conscientiousness and emotional stability scores to drift up over the next decade whatever you do about them.