not on the bias
I did some experimenting and my answer is, Yes. And . . . No.
see how it's all bunchy?
As you can see in these first two pictures, I did not cut the fabric on the bias because I didn't have enough of the solid. I thought I would make do with the little that I had. It didn't work and I ended up ripping off the trim stitch by stitch.
So when adding bias tape to any type of curve, you really do need to cut the fabric on the bias. I ended up buying some fabric and using the diagonal line of my cutting mat to make that perfect 45 degree angle.
on the bias
It turned out much better.
But I had to buy a 1/2 yard of fabric to get nice long strips (I don't like lots of seams on my bias tape). I needed about 3 strips for this project and had two bizarre triangle shaped pieces of fabric left over. What a waste. If you're like me and you rarely use bias tape around a curve, I recommend that you just buy a pack of bias tape in a color that's close enough. You'll get about 3 yards for less than $3. That's much less than the cost of 1/2 yard of nice quilt-weight cotton and you won't have fabric left over -- just extra tape to keep on hand for future projects.
If you're using the bias tape for a hem (even a sleeve cuff), not cutting on the bias is going to work out just fine and it won't waste so much fabric.
Edit: One of my friends just asked me what bias tape is. I forgot that not everyone who reads this sews! My plan wasn't to make Project: Project all about sewing but the weather has had other ideas and I've done little else since January.
Anyway, bias tape is that orange stuff in the pictures. It is basically 2" strips of fabric with three folds that create a clean little way to cover up the unsightly edges of a project. It is usually used for binding the edges of quilts. Quilt bindings need to be cut on the bias so that one single thread of a piece of fabric isn't the one getting all the wear. If the fabric is cut on the bias, or the diagonal, then many threads make up the edge, prolonging the life of the binding. Quilt binding does not need to be cut on a true bias (the 45 degree angle); it just shouldn't be cut on the square (or parallel with either direction of the weave of the fibers).
For clothing purposes, I cut my bias for the experiment on the true bias so that it would have the maximum give and stretch. But like I learned, if you're just adding some trim to a cuff or a hem, any way you cut it is going to work.
Hope this helps!