Fly threads come in different degrees of thickness. For large flies (large streamers) you may use 3/0 or 6/0, but when you tie the typical trout/grayling patterns you should strive to move toward 8/0 or smaller. Adjust your tension while binding the materials and be vigilant to not nick the hook point while wrapping the thread. This often happens while attaching materials to the rear portion of the fly. Learn to maneuver the bobbin to avoid that hook point because that is where the thread will often break.
I have taught fly tying and often had a few spare bobbins ready to go with thread, for when students invariably broke their thread. I would instruct them to pinch off the thread on the hook so it would not further unravel and hand them the the backup bobbin. A few wraps over the existing thread on the hook would bind it down and the tying of the pattern could continue. A backup bobbin ready to go is not a bad idea.
Also, threads are often flat like a ribbon but have a twist imparted to them at the factory (clockwise) that is further induced while tying. This creates a kink in the thread that is often evident as slack is allowed in the thread (the loop of thread jumps off target). Letting the thread hang and imparting a gentle counter clockwise spin to the bobbin will eventually return the thread to a flat state. In the flat state, the thread is more manageable. To be honest here, I spent a good amount of time oblivious to this years ago because it was never mentioned to me. Only later did I learn about the twist. I recall letting my bobbin dangle beneath the vise while preparing materials and seeing the bobbin twist.