Let be introduced the Sobolev-type inner product \[ (f,g)=\frac12ıt^1_{-1}f(x)g(x)dx+M[f'(1)g'(1)+f'(-1)g'(-1)], \] where $M\geq0$. In this paper we will prove that for $1\leq p\leq\frac43$ there are functions $f\in L^p([-1,1])$ whose Fourier expansion in terms of the orthonormal polynomials with respect to the above Sobolev inner product are divergent almost everywhere on $[-1,1]$. We also show that, for some values of $\delta$, there are functions whose Legendre-Sobolev expansions have almost everywhere divergent Cesáro means of order $\delta$.