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: 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.

This is very often false. For example, take a line $L$ in $\mathbb P^3$, and two general surfaces of degree $d > 1$ passing through $L$. Their intersection is the union of $L$ with an irreducible curve $C$ of degree $d^2-1$. Suppose that $C$ is cut by two surfaces $S_1$ and $S_2$ of degrees $d_1$ and $d_2$ at most $d$; call $m$ the intersection multiplicity. Then $d^2 - 1 = d_1d_2/m ≤ d_1d_2 ≤ d^2$. If either $d_1$ or $d_2$ is less than $d$, then $d_1d_2 ≤ d^2 -d$, and this is impossible; so $d = d_1 = d_2$, which implies $d^2 - 1 = d^2/m$, and this is impossible.

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 $d$.