Edit: the first version of this was completely wrong, I hope this one works.

Take a line $L$ in $\mathbb P^3$, and two general surfaces of degree $d > 3$ passing through $L$; it is not hard to see that the only line contained in $S_1$ is $L$. Their intersection is the union of $L$ with an irreducible curve $C$ of degree $d^2-1$. Now, suppose that $S$ is a surface of degree $d$ containing $C$; then the intersection of $S$ with $S_1$ must be the union of $C$ and a line, which must coincide with $L$. So every surface of degree $d$ that contains $C$ also contains $L$, and this means that we can't cut $C$ with surfaces of degree $d$.

[Edit]: this also works for $d = 3$. A smooth cubic surface famously contains 27 lines, but they define different classes in the Picard group.