My understanding is Atiyah-Patodi-Singer solved the index theorem for manifold with boundary only for certain types of Dirac operators, correct?
There is still no (or no hope to get) uniform theorem for the Dirac operator associated with any Dirac bundle (in the sense of Gromov-Lawson)? 



Is there any general index theorem for manifold with boundary?
1. The APS theorem works for any Dirac-type operator; see e.g. the excellent monograph by Booss-Wojchiecowski on this topic.
2. More than four decades ago, Boutet de Monvel has described a general set-up for dealing with boundary value problems that mimicks the K-theoretic approach to the index theorem on closed manifolds. For a modern presentation of this point of view I recommend this paper by Melo-Shrohe-Schick [arXiv: 1203.5649](https://arxiv.org/abs/1203.5649) and the references therein. It involves some noncommutative geometry because the symbols in the Boutet-de-Monvel calculus of elliptic boundary value problems define elements in the $K$-theory of a noncommutative $C^\*$-algebra. In the case of closed manifolds symbols of elliptic operators lead to elements in the $K$-theory of a *commutative* $C^\*$-algebra.


