Suppose that I try to explain $Y$ with $X_1$ and run a simple regression $y_i = \beta_0 + \beta_1x_{1,i}+e_i$. The result is that $\beta_1$ is not significant, so we conclude that $X_1$ does not explain $Y$.
Then I add another independent variable, $X_2$, in the regression. This time, $\beta_1$ is significant but $\beta_2$ is not(!). Is this kind of a situation possible or perhaps have I made a mistake? If it is possible, what is the interpretation? What could explain the observation that $X_1$ alone does not explain $Y$, but with an additional variable $X_2$, $X_1$ becomes meaningful, yet $X_2$ itself is not explaining $Y$?
