On page 188 of "Topology and Analysis," Booss and Bleecker claim that the following theorem does not in general carry over to the case when the manifold $X$ has a nonempty boundary:
Let $E$ and $F$ be vector bundles over $X$, $P\in\mathrm{Ell}_k(E,F)$, and $s,s-k\ge 0$. Then
(a) The extension $P_s:W^s(E)\to W^{s-k}(F)$ is Fredholm with index independent of $s$.
(b) $P^*$ is elliptic and $\operatorname{coker}P_s\cong\ker (P^*)_{s-k}$.
(c) $\ker P_s=\ker P$.
(d) $\operatorname{index}P=\operatorname{index}P_s$ depends only on the homotopy class of $\sigma(P)$ in $\mathrm{Iso}_{SX}^\infty(E,F)$.
For a different result, they specify that part (a) of that result carries over. That makes me think they are implying no part of this result carries over. But the only results we really need for this are Sobolev embedding and Rellich's theorem, both of which hold on manifolds with boundary. The only part of I'm not sure of is the independence of the index from $s$, because the argument given in the book is very vague. But one can circumvent it using regularity theory (i.e. prove (b) and (c) first), as is done in Gilkey, Hörmander, or Ward.