The Bleeding Heart plant (Dicentra spp) has delicate stems with a tendency to break easily, so whether you can start a new plant from an existing one isn't an uncommon question. The answer depends on whether the stem is broken or the entire plant has broken away from the root.
What to Do With a Broken Heart
If the plant has broken off a stem, the only thing you can do with it is to throw the stem away and don't worry about it. Bleeding hearts are hardy, dispite how fragile they look. A broken branch or three won't hurt the plant at all.
More Serious Damage
If your Bleeding Heart has had a really bad accident and some of the stems are ripped away from the root, you can still salvage the plant -- and have a new one. As long as the branch that ripped off has some root attached, you can quickly plant it in a new location and can fully expect it to grow. Cover the exposed root on the original plant, and it probably won't even wilt.
For whatever reason, your Bleeding Heart has been mauled. Don't panic. You will not be Bleeding Heart-less next spring. Save everything that looks like a piece of the root, which looks and feels spongy, and either bury them where you want new plants to grow or place parts of the root in potting soil in pots. Keep them in partial shade and water only when the dirt feels dry. Watering too frequently might rot the root.
You may not see growth from a root cutting for a few weeks, so don't give up. It also takes a few seasons for the plant to reach its full size.