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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.