Suppose $X=X_1 + X_2$ and $Y=Y_1 + Y_2$ are independent random variables such that $X_1,X_2$ are independent and $Y_1,Y_2$ are independent. Does this imply that $X_i,Y_j$ (for $i,j\in \{1,2\}$) are independent?

I'm asking this question because I'm a little confused about the accepted answer for Sum of two independent binomial variables; in the last part of the proof, it's clear that the $B_i(i=1,\ldots,n+m)$ are binomially distributed, but it's not apparent to me why they are all independent (which somehow follows from the fact that $B_1,\ldots, B_n$ are independent, $B_{n+1}, \ldots, B_{n+m}$ are independent, and $X=B_1+\cdots+B_n,Y=B_{n+1}+\cdots+B_{n+m}$ are independent).

Edit: Added some motivation.