The additive assumption means the effect of changes in a predictor on a response is independent of the effect(s) of changes in other predictor(s).
However, with regression, say, with one continuous predictor and a continuous response, the addition of an additional predictor variables can impact the relationship between a predictor and the response.
My naive interpretation of additivity would suggest that the addition of additional predictors does not change the relationship between, say, the first predictor and the response.
Why is this not the case? What exactly does the additive assumption mean?