If the edges of a graph can be partitioned into two planar graphs, it has thickness 2 (a biplanar graph). This Demonstration shows some thickness-2 embeddings of various quartic graphs. In the first step a Hamiltonian cycle is chosen. One of the graphs, the icosidodecahedral graph (26), is planar and actually has thickness 1.

Non-planar quartic graphs have geometric thickness 2 [1], and therefore have thickness 2.

[1] Christian A. Duncan, David Eppstein and Stephen G. Kobourov, "The Geometric Thickness of Low Degree Graphs", 24 Dec 2003. http://arxiv.org/abs/cs/0312056.