For two languages over the same alphabet that are not context-free, can their intersection be context-free, Or does at least one of them have to be context-free?
The intersection of two CFL may or may not be a CFL. We can show that both possibilities exist. For example if both $L_1$ and $L_2$ are regular then $L_1∩ L_2$ is regular and therefore CF.
For intersection of non-context-free languages be context-free, you can assume that the intersection could be empty.