multicollinearity and significance

[maavtc]

I'm doing a regression with a couple of highly correlated variables. I don't really care about the significance of individual estimators because I'm only using them to build an index. As it is often the case when there is multicolinearity, i get a estimator with the wrong sign which is not a problem. The thing is I wasn't expecting that the estimators would be statistically significant. Does that mean that there is a problem with my variance estimation or is it possible to find statistically significant coefficients in my multiple regression and have completely different coefficients when I regress each variable individually?
I'm asking the questions because I don't want that people see my regression output and try to interpret the coefficients even if they are statistically significant.

Thank you,

[startz]

is it possible to find statistically significant coefficients in my multiple regression and have completely different coefficients when I regress each variable individually?

Entirely possible. However, it will take some arguing to convince anyone that the single variable coefficients are preferred to the multiple regression.