What happens to the poles of a bar magnet if it isn't completely broken in two, but only partially broken? At what point does it become two magnets instead of one? 
 A: Magnet is said to split into two parts when the magnetic circuit is broken.In your case if the magnet is partially broken there is a fringing flux in the air gap but most of the field lines pass through the low reluctance path,which in the above case is the unbroken part.And if there has to be fringing of flux there has to be two unlike poles nearby and hence two poles are created as soon as the magnet is broken.Which may be very weak as most of the flux flows through the unbroken part.
A: A magnet is not a single object. Internally it is made up of microscopic magetic domains that are in effect tiny individual bar magnets. The large magnet you hold in your hand is a composite of all the tiny magnetic domains.
So when you talk about breaking a magnet you aren't really breaking any magnets, just separating the domains. The result might look something like:

For the unbroken magnet the fields of all the individual domains add up to give nice clear North and South poles. For the partially broken magnet there will still be clear poles at the ends, but near the break the fields of the individual domains won't add up in a tidy way and the field will be somewhat messy. As you complete the break and pull the two halves apart the field near the break will settle down to give a new North and South pole at the broken ends.
