Let $A$ be a graded decreasing filtered chain complex, with cohomology differential $d$. Let $^nA^p=A^{p,n-p}$ be the elements of $A$ with total degree $n$, filter degree $p$, (and therefore with complementary degree $n-p$).
In Deligne(Theorie de Hodge II Prop 1.3.2), there is a proposition that a spectral sequence associated to a filtered complex is degenerate, i.e. $E_1^{p,q}=E_\infty^{p,q}$, iff $A^p \cap d(^{n-1}A)=d(^{n-1}A^p)$.
This seems dubious: $LHS=B_\infty^{p,n-p}$ and $RHS=B_0^{p,n-p}$. $LHS=RHS$ iff $B_\infty^{p,n-p}=B_0^{p,n-p}$. I don't see how this implies that $C_\infty^{p,n-p}=C_0^{p,n-p}$.