Imagine the box is not there, but that the center line was trimmed back to where the box would be. Is there a way to easily do this in Solidworks? I think in another software companies Express Tools it was called EXTRIM. You could draw a box, circle, whatever and quickly clean away lines around the object you created.
I don't know if this setting will help or not. It appears from your picture that the dimension is somewhere within the part perimeter. It is generally better practice to use a leader line and move it out from the part to avoid confusion.
The only place I see it working in SW is hatched cross section views -
Thanks for the input gentlemen. Jeff Holliday I agree that it's poor drafting practice (or at least what I had been taught) but this company likes to set it's drawings up this way.
Believe me - we all can understand that
You can do a little work around, if you need to have the text right there... In the part file create a new configuration, call it drawing info or something, then draw a small rectangle (or an odd shape), cut extrude thru, then in the drawing load that configuration and put your text in the box and select the outside of the box convert the edge entities to lines and then change the line color from black to the same color as the paper color (white) then when it prints, it will skip over those lines and now you have what you're looking for, but additional work........
Here is a rough and ugly workaround;
Activate the view you want to modify, I.E RMC lock view focus or double click inside view
Sketch two horizontal lines, one above the other, around the size of your dimension in the area you want to block out. Image 01
Change the line thickness and color of these two lines. Color to white and thickness to .156. Image 02 and 03
The drawing it will look ugly on your computer screen. Image 04 but it will print the way you want. Image 05. You might need to make the lines closer together and or add a third line. Also center lines will still bleed through so you will need to move the end point of the center line to one side of the dimension and add another one to the other side to complete.
With center lines adjusted, Image 06
This was my work around. Would be nice if you could trim text like you can in ProE (or is it Creo now...)
Found this work around with hatch:
If you create a solid hatch, you can then select Line Color 
from the Line Format toolbar. Works for what they want.
Found this work around with hatch:
If you create a solid hatch, you can then select Line Color
from the Line Format toolbar. Works for what they want.