The following paper is relevant:

> *Finite coverings by normal subroups* by Brodie, Chamberlain and Kapp, PAMS 1988.

The main focus is on coverings of infinite groups although their main theorem is still interesting in the finite case.

> **Theorem**. A group
has a nontrivial finite
covering by normal subgroups if and only if it has a quotient isomorphic to an elementary abelian $p$-group of rank two for some prime $p$. 

One of the corollaries to this theorem is also relevant for finite groups:

> **Corollary**. Let $G=\bigcup\limits_{i=1}^n N_i$ where $N_1,\dots, N_n$
form an irredundant covering of $G$ by proper normal subgroups. Then $G/D$ with $D=\bigcap\limits_{i=1}^n N_i$ is finite and solvable.

This corollary effectively reduces the question of coverings by normal subgroups to the study of solvable groups. The paper can be viewed [here][1]. If you don't have access email me and I'll send it to you.

  [1]: http://www.ams.org/journals/proc/1988-104-03/S0002-9939-1988-0964841-9/S0002-9939-1988-0964841-9.pdf