The backyard fence is like a canvas and you are the painter.
No doubt, you can paint the fence any color.
How about decorating the fence as a focal point in the yard?
You can hang a large mirror up or several smaller ones creating an illusion of depth in a small space. Placing picture frames or fancy plates on the fence for an interesting aspect. Add a few climbing plants like clematis to finish the look.
Use an ornamental trellis or bed frame for those climbing plants. Everything will stand out better against a light-colored fence.
If you are short on growing area, try hanging pots or planters, or building boxes for plants onto the fence. Try nailing open wooden crates to the fence with pots and plants in them.
Do not forget to experiment with lighting. Try placing solar deck or spotlights around for a stylish effect playing off of focal points and shadows. It also adds to the security of the backyard at night.
If you are an artist, you can always paint a mural on your fence.