Suppose $X$ is a general smooth hypersurface of degree $\ge 6$ and $Y$ be an irreducible hypersurface of degree $\ge 2$. Let $X \cap Y$ has at least $5$ nodes. Is it possible that $4$ nodes of $X \cap Y$ lie on a plane ?
I guess not.
The reason i have in my mind is the following: If possible let $H$ be a plane containing $4$ nodes. In this case $H$ is itself tangent to $X$ at these $4$ points which is a contradiction as a plane can be tangent at at most $3$ points. Please correct me if i am wrong.
