Thanks for your enlightening comments. Regarding the examples of a predictor variable being significant in univariate but not multivariate (or vice versa), what then is the point then of having univariate analyses? Especially given that the more 'real' situation is described by the multivariate case (and rarely will there be only one possible predictor). At worst, univariate analyses could be misleading (see Masteras's example of heart weight of mice above).
Also, could you answer one of my questions from above:
1. In case2 where the independent variable ('X') is significant in both univariate and mutlivariate cases, what does the difference in the R-square of the univariate analysis and the partial R-square for 'X' of the multivariate analysis represent? Would it be the variance explained common to both 'X' and 'Z'?