The first thing to do is to check the Tools>Check and find the minimum radius of the part.
You can't tell much about the structure of a part from an eDrawing, but you also need to find out what a "degenerate surface" is, and try to eliminate degeneracies on the part.
This is the type of problem that vexes even experienced users.
You could manually thicken it, by offsetting individual faces, then bridging the gap.
There are probably half a dozen ways to work around the problem, and several ways to fix the underlying problem so the thicken works correctly. It's difficult to solve a problem like this without being able to play with the actual part.
If I had to guess I would say that the problem is probably at the narrow ends of the part where the mirrored halves come together. Artifacts on the edrawing make it look like those points where all the surfaces come together are degenerate points (underlying U-V directions come down to a single point), and the curvature around those points on a microscopic level is probably irregular and will probably yield a very small curvature which prevents a surface from offsetting to make the thickness.
To get rid of those, either trim out the area around the points and use a Fill surface to fill it in, or if you have the original SolidWorks data, go back and make the surfaces a different way that avoids the points.
Get yourself a copy of the SolidWorks Surfacing Bible to learn more about this type of problem. http://www.dezignstuff.com/blog/?page_id=618