If the thread looks new and clean and the bobbin is over 1/2 full I'll hang on to it and try to use it in a quilt or a whacky bag. However if the bobbins are old looking I peal off the thread and trash it. Many times I've put what I thought was a full bobbin in and right quick run out of bobbin. Did it break, no it was a layer over another color, sometimes many colors. Recently I handed my wife a bobbin with light brown thread in it. She instantly started having tension trouble and thread breakage. The thread and bobbin looked good ... from the outside. Under the thread was rust. It had glued the thread together and that caused a lot of trouble.
Now I just peal them off and toss the thread in the trash. It's not worth the trouble.
If I fill a bobbin and don't use all of it, I'll put the bobbin in a pill bottle with the spool of thread it came off. Then I can use it when I need that thread again.
Joe