Edit: New information has been posted, check July 2014
I've come across this countless times now, I will confirm this. The SMALL mobs with names above their heads (Mini-bosses) those can be tamed into pets, but due to their status they have a roll to immediately become Rare right away, and if not they have a second chance when they reach your level for the first time they'll have the same roll again.
HOWEVER, no matter your feelings for Puppy the first mini-boss. Do NOT go back and tame it. There is a glitch where you will pick up his tail a second time (There were two quests for his tail in the first town, not sure why but you have to kill him twice while having those quests on hand.) and you'll lose a slot in your Etc. bag, same can be done with some bosses and other quests.
Only difference between taming a mini-boss and a normal mob is the mobs health, defence, strength and what not. Which the mini-boss can actually kill you before you tame it, thankfully you can pot your health back while taming. Though another issue with mini-bosses is their minions, they will endlessly spawn (within the normal spawning times) and become aggressive against you, so watch out for them as well.
Also a piece of Pokemon advice, weaken the mob's hp to around 1/3 or 1/4. Most of the time they will run away with a skull above their head. At this time you can take advantage of them and try taming, this will give you some time for non-combat taming compared to being attacked while taming them.