When I was your age Coyotecatcher we used to hunt animals with out BB guns everynight! The answer to your question depends on the type of BB gun that you own. My brothers and I have taken several Coons, rabbits, foxes, and other small game of the sort with ours. My brothers BB gun was a rather powerful one however. Comparable to a .22 caliber rifle. Mine was just the original 10 pump air rifle, and it did do the job a number of times on the smaller game such as squirrel, coons, and some p-dogs.
Remember that you need to think of the animal first. If you don't think that you can place that shot in a very vital area, let the little critter live. We past up several little animals due to a bad shot. If you are going to attempt to hunt with your bb gun, be a responsible hunter, and be considerate to the animal.
Looking back, if I had the option to hunt with a 410 shotgun vs. my BB gun for fox, and Coon I would take the 410....
Good hunting to ya!