Lars-Erik,
It looks to me like your shape is not properly-closed geometry. I bet if you open the ShapeSheet, you'll see lots and lots of Geometry sections, as opposed to one single Geometry section that ends where it starts (ie: closed)
Sometimes you can get Visio to create a single outline by selecting all of the bits, then choosing:
Shape > Operations > Fragment
(Vorm > Verrichtingen> Besnoeiing/Plak) ..stop laughing!
If the line segments are close enough, you might get lucky and Visio will cut out a single shape that you can properly fill.