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 Frobenius's theorem, if $G$ is a finite simple group, and $M$ is a nilpotent Hall subgroup of $G$ disojoint from its other conjugates, then $M$ can't be self normalizing (there are finite simple groups which have a Sylow $2$-subgroup which is a maximal subgroup, eg ${\rm PSL}(2,17)$). Hence it follows for general reasons(without CFSG but as Nick says, with some quite 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 and disjoint from their other conjugates ( apart from the identity). However, at the moment I don't see how to tackle directly ( ie without CFSG, and with easier results than Nick quotes ) partitions of the type asked about here (even just for simple groups).

Later edit: In a completely different direction, (though this may be covered by Baer's work, which predates Carter subgroups), a finite solvable group $G$ is never a union of proper nilpotent self-normalizing subgroups : for $G$ has a unique conjugacy class of nilpotent self-normalizing subgroups, the Carter subgroups, and no finite group is the union of a single conjugacy class of proper subgroups (if $G$ itself is nilpotent, it has no proper self-normalizing subgroups at all).