Statistically significant variable doesn't always necessarily show any impact it may bring upon to others.
A statistically insignificant variable, on the other hand, may bring upon a huge impact to its environment.
But whichever may be true, we might identify the variables wrongly. The error in the estimation is, unfortunately, another variable which we might have missed to identify.
A result, eventually, will only be as good as the model itself.
Theoretical experiments could only tell you so much.