To every cubic hypersurface $X$ we associate the parameter space of lines contained in $X$; this is called the Fano scheme of lines on $X$ and denoted $F(X)$. If X admits an isolated singular point of ADE-type, we prove that $F(X)$ admits hypersurface singularities of the same type transversally along the regular part of its singular locus. As was shown by H. Clemens and P. Griffiths, the Albanese variety $\operatorname{Alb}(F(X))$ of the Fano scheme of lines $F(X)$ on a smooth cubic threefold $X$ is isomorphic to the intermediate Jacobian $IJ(X)$ of $X$. G. van der Geer and A. Kouvidakis generalised this result to nodal cubic threefolds, replacing the Albanese variety $\operatorname{Alb}(F(X))$ by the Picard scheme $\operatorname{Pic}^0(F(X))$.We study more generally degenerations of the Picard scheme $\operatorname{Pic}^0(F(X))$ when the smooth cubic threefold $X$ degenerates to a cubic threefold with unique singular point of type $A_k$ and prove that these degenerations define points in Mumford’s partial compactification $\mathcal{A}_5'$ of the moduli space $\mathcal{A}_5$ of principally polarised Abelian varieties of genus five.