What do you mean the lines don't have nodes?
You can't know if a mesh is good or not by look of it. You need to do sensitivity analysis for the mesh it to make sure it is sufficiently fine. Although, if the physics if merely about conduction through a solid, often a relatively coarse mesh is enough. But you still need mesh sensitivity study as described in this tutorial. You can use mesh adaptation, or decide to refine the meshes manually with judgment. I choose the latter.
What John is actully seeing is a visual bleeding of the mesh going through the walls. Basicallly, he is seeing some of the element definition on the top surface on the side walls. Nothing to be concerned about.
You can control the amount of visual depth or bleeding in the Z-clipping window (view ribbon) and then adjusting the mesh factor value to a lower setting. This can make very skinny heat sinks look a lot better when you turn on the solid mesh view.