No, but if the codimension of the two submanifolds is high enough, then it is possible. Generically the intersection $A$ will be a submanifold of dimension $\dim(A)=\dim M+\dim N-\dim X$. Hence if the dimension if $X$ is big enough the intersection will be empty.

No, but if the codimension of the two submanifolds is high enough, then it is possible. Generically the intersection $A$ will be a submanifold of dimension $\dim(A)=\dim M+\dim N-\dim X$. Hence if the dimension of $X$ is big enough the intersection will be empty.