When we find solutions to a PDE through separation of variables are we guaranteed to get all possible solutions to the original PDE or could there be missing solutions where the variables are not separable?

For example: PDE P(x,y,z) assumed separable and we find a linear set of solutions in the form X(x)Y(y)Z(z). Could there be a solution in the form A(x,y)Z(z) where the variables x and y are not separable that we can not create through a linear combination of solutions in the form X(x)Y(y)Z(z)?