Suppose we have a multiple regression with $n$ predictors. All of them have significant correlations with the response except for one. Could the predictor with the zero correlation to the response be significant?
The prototypical situation in which a variable has a zero marginal correlation with the response, but a significantly non-zero association conditional on the inclusion of the other covariates is called suppression. There are a number of threads on this topic on CV:
- Suppression effect in regression: definition and visual explanation/depiction
- X and Y are not correlated, but X is significant predictor of Y in multiple regression. What does it mean?
- How can including an IV uncorrelated with the DV improve a multiple regression model?
It is also possible that the variable is not a suppressor, but is relevant and is just 'cancelled out' by another variable with the opposite effect that it is correlated with. Here are some things to read to help understand that:
Yes, it can happen and even more extreme scenarios might happen.
Think that you have 2 features, the beginning of a period $s$ and its end $e$. Assume the the concept is the period length $l = (e-s)$ and we will build the data set so both $s$ and $e$ are independent of $l$.
Correlation consider only a single feature and the concept and it will be $0$ (by construction). However, given the start and the end of the period, you have all the information on its length.
Returning to your question, you can build a scenario in which the zero correlation predictor and on of the others predictor sum is the concept. Multiple regression is very suitable to these problems so you can get perfect prediction. That will be even easier if you'll remove all the rest of the predictors.