There's a specific term for reusing ingredients for stock twice: remouillage (which literally means a "rewetting"). Usually this "second stock" is not used directly for broth, as it has significantly less flavor than the primary stock. That said, depending on the type of bones, the amount of meat used in making the stock, etc., it may still have a very pleasant (if lighter) flavor.
In traditional French cuisine it tends to be used as a cooking liquid to make a new stock with (that is, you might cook chicken #1 twice, and use the second stock from chicken #1 to make a richer stock using the bones from chicken #2). It can also be reduced for a glace, in which case the significant concentration will make it taste a bit more flavorful.
I personally tend to do this frequently when I'm making stock and save the "second stock" for miscellaneous uses, like a cooking liquid for rice or vegetables, or as the basis for a future broth.
Generally, doing a third (or more) use of the bones for broth will extract very little flavor, mostly only giving you a bit of the remaining gelatin. Any flavor that does still exist will also become increasingly unbalanced.