To add to DVK and Mac Cooper's answers:
In Fantastic Beasts & Where to Find Them; there is a section regarding Controls on Selling and Breeding which states:
The 1965 Ban on Experimental Breeding has made the creation of new species illegal.
-Fantastic Beasts & Where to Find Them, Page xix.
However, there is a scribble (Harry or Ron's) saying "but no one's told Hagrid". I'm assuming they are implying Hagrid did breed experimental creatures; which perhaps could have been the Blast-Ended Skrewts.
I have attached a scan of the page as well (hope this is ok).
Another part of this theory is that the Blast-Ended Skrewts don't appear in Fantastic Beasts & Where to Find Them, but rather seem to be an amalgamation of a Fire Crab (Shoots fire from its read end when attacked) and a Manticore (Has a scorpions tail and repels most charms) and maybe some other creature(s?), perhaps a Flobberworm.
All throughout Goblet of Fire Hagrid seems to be learning just as much as the students do about these creatures to the point where he doesn't even know what they eat, which you would think that the Care of Magical Creatures teacher would at least know that before bringing them to class:
Yer jus’ feedin’ ’em today. Now, yeh’ll wan’ ter try ’em on a few
diff ’rent things — I’ve never had ’em before, not sure what they’ll
go fer — I got ant eggs an’ frog livers an’ a bit o’ grass snake — just
try ’em out with a bit of each.”
-Harry Potter and the Goblet of Fire, Chapter Thirteen (Mad-Eye Moody).