I realize this is a (very) dated thread, but since one of my colleagues asked me this very same question this week and finding nothing on the Web that I could point him to, I thought I would add my two cents "for posterity" here. I am not convinced that the answers provided to date answer the OP's question.
I am going to simplify the problem to involve only two independent variables;
it is very straight-forward to extend it to more than two. Consider the following scenario: two independent variables (X1 and X2), a dependent variable (Y), 1000 observations, the two independent variables are highly correlated with each other (r=.99), and each independent variable is correlated with the dependent variable (r=.60). Without loss of generality, standardize all variables to a mean of zero and a standard deviation of one, so the intercept term will be zero in each of the regressions.
Running a simple linear regression of Y on X1 will produce an r-squared of .36 and a b1 value of 0.6. Similarly, running a simple linear regression of Y on X2 will produce an r-squared of .36 and a b1 value of 0.6.
Running a multiple regression of Y on X1 and X2 will produce an r-squared of just a wee bit higher than .36, and both b1 and b2 take on the value of 0.3. Thus, the shared variation in Y is captured in BOTH b1 and b2 (equally).
I think the OP may have made a false (but totally understandable) assumption: namely, that as X1 and X2 come closer and closer to being perfectly correlated, their b-values in the multiple regression equation come closer and closer to ZERO. That is not the case. In fact, when X1 and X2 come closer and closer to being perfectly correlated, their b-values in the multiple regression come closer and closer to HALF of the b-value in the simple linear regression of either one of them. However, as X1 and X2 come closer and closer to being perfectly correlated, the STANDARD ERROR of b1 and b2 moves closer and
closer to infinity, so the t-values converge on zero. So, the t-values
will converge on zero (i.e., no UNIQUE linear relationship between either X1 and Y or X2 and Y), but the b-values converge to half the value of
the b-values in the simple linear regression.
So, the answer to the OP's question is that, as the correlation between X1 and X2 approaches unity, EACH of the partial slope coefficients approaches contributing equally to the prediction of the Y value, even though neither independent variable offers any UNIQUE explanation of the dependent variable.
If you wish to check this empirically, generate a fabricated dataset (...I used a SAS macro named Corr2Data.sas ...) which has the characteristics described above. Check out the b values, the standard errors, and the t-values: you will find that they are exactly as described here.
HTH // Phil