Suppose I set up a multiple regression with one continuous variable $X_1$, one (2-level) categorical variable $X_2$, and their interaction $X_1 X_2$.
This will result in four parameter estimates ($\beta_0, \beta_1, \beta_2, \beta_3$) and two different lines with different slopes ($\beta _0$ versus $\beta_0 + \beta_2$) and intercepts ($\beta_1$ versus $\beta_1 + \beta_3$), one line for each value of the categorical variable.
Now, could I deduce from the p-values in the model output that I. the slope for the line corresponding to the baseline value of the categorical variable is not statistically significant, but II. the slope for the line corresponding to the other value of the categorical variable is statistically significant?
I think not, because the slope estimates for each line are intertwined. But I am not sure.