In running a regression model with three variables, two of the three come back as statistically significant at the 95% level of confidence. The one that doesn't is the intercept - so is my model still valid?
In general, I would say yes, the model is still valid. Your equation would have a problem if the underlying model you are trying to estimate expects a non-zero itercept term.
However, for most of the model's I estimate, I have no a-priori notion or expectation of what the intercept ought to be. I am much more interested in the slope term.