To expand a little on Nick Gill's answer: it was a common theme in some early classification-type theorems that a simple group might be a disjoint union of Hall subgroups with trivial intersection of any two of them. This occurs for example, in Suzuki's classification of simple CA-groups ( groups in which all non-identity elements have Abelian centralizers)- I do not know whether Suzuki was thinking about groups with a partition before he did that classification. The Suzuki groups are simple CN-groups ( nonidentity elements have nilpotent centralizers)-there are simple CN-groups with no such partition, eg ${\rm PSL}(2,7), {\rm PSL}(2,9), {\rm PSL}(2,17)$ - in these last groups, Sylow $2$-subgroups can have non-trivial intersections. Quite often, delicate character theory was needed in these classifications. Also, by other normal p-complement/transfer theorems, if $G$ is a finite simple group, and $M$ is a self-normalizing nilpotent Hall subgroup of $G,$ then $M$ must be a $2$-group (essentially a result of J.G. Thompso- and there are cases where a $2$-group occurs, eg in ${\rm PSL}(2,17)$. Hence it follows for general reasons(without CFSG but as Nick says, with some difficult group theory) that in a finite non-Abelian simple group which admits a partition by nilpotent Hall subgroups, these can't all be self-normalizing. However, at the moment I don't see how to tackle directly ( ie without CFSG) partitions of the type asked about here (even just for simple groups).