Let $X$ be a proper and smooth scheme over $\mathbf{C}$ and let $\mathbb{L}$ be a local system of finite dimensional $\mathbf{C}$-vector spaces. By the Riemann Hilbert correspondence, to $\mathbb{L}$ one can associate a locally free sheaf $\mathcal{F}$ of $\mathcal{O}_X$-modules with an integral connection $\nabla \colon \mathcal{F} \to \mathcal{F} \otimes \Omega^1_X$ such that $\mathbb{L} = \mathcal{F}^{\nabla = 0}$. In particular the de rham complex of $\mathcal{F}$: $$ 0 \to \mathcal{F} \to \mathcal{F} \otimes \Omega^1_X \to \cdots $$ gives a resolution of $\mathbb{L}$. This resolution is not injective, but still we can use hypercohomology of the complex to compute the cohomology of $\mathbb{L}$, and we get a spectral sequence of hypercohomology.

If $\mathbb{L} = \mathbf{C}$ is constant, this is the usual Hodge to de Rham spectral sequence, that degenerates, immediately. What can be said in general? I guess that it not true that the sequence always degenerates. (But for example it should be true if $\mathbb{L} = f_\ast \mathbf{C}$ for a proper and smooth morphism $f \colon Y \to X$.)