I'll restate this to ensure we're addressing the same issue: There's regressors A and B which themselves are correlated. The result of the regression is that neither A nor B show an effect. Yet when A is corrected for B or vice versa, then there is an effect. This is known issue when regressors are confounded; the explanation has to do with the nature of the t-test for betas or contrasts. A nice exposition of the issue can be found in Andrade et al. (“Ambiguous Results in Functional Neuroimaging Data Analysis Due to Covariate Correlation”, NeuroImage 10, 483–486 (1999)), at e.g. http://people.hnl.bcm.tmc.edu/jli/reference/209.pdf A more general exposition is given in _ Applied Linear Statistical Models_, Neter et al., 4th ed. in the case of correlated explanatory variables. In summary, if you do a t-test on each beta separately, you might find none of them significant, but if you test all of them together with an F-test, they can be highly significant (together).