Re: Breaking up a variable
If you have two sub-variables you have somet cohing like B1*X1 + B2*X2 and with one sub-variable I'm assuming its B3*(X1+X2) = B3*X1 + B3*X2 for appropria scete co-efficients B1, B2, and B3.
In the first scenario you are testing B1 = B2 = 0 for non-significant but in the second you are testing B3 = 0.
If we consider a situation where test statistic for B1 = -1.5 and B2 = -1.5 then both will be in-significant but if B3 = -1.5 + -1.5 = -3.0 (assuming that can happen) then this will be significant.
For more precise analyses you should consider the covariance between X1 and X2 and this relationship on the response variable.