Use the right bullet and you will not have any problems. I have two killed two black bears now with the 64 grain power point (150 lbs at 150 yards) and 225 lbs (10 feet straight on) with a 22 250 at 3260 fps. Both bullets exited with massive internal damage. I ran the 64 grain power point versus the hornady 60 grain SP, hornady 55 grain SP, Speer 70 grain, 63 grain Sierra SMP and the 55 grain Sierra SMP into dry phone books on two different occasions. Dry phonebooks or newspaper will really stress the bullet and a lot of gun writers (John Barsness in particular) feel that is gives a good simulation of what a bullet will do when it hits bone. The 64 grain PP was the best of the bunch, each penetrating an extra 20 % over the others, although the 60 grain Hornady SP (without the cannelure) was a pretty close second. Depending on the twist in your particular rifle, the longer 53 grain TSX is another super penentrator although I find in my 1/14 twist that anything over 0.8 inch in bullet length starts to really ran the ragged edge of bullet stability. The 60 grain Nosler partition is another very good bullet. My 22 250 will hold them 5 shots for 1.25 inch at 100 meters and based on the penetration tests versus my 308 WIN 150 grain hornady SP and 165 grain BTSP, they will penentrate on par with much larger calibers. Of course, since the .224 bullet is much smaller in cross sectional diameter, the overall wound volume at equal penetration will not be as large. Intuitively, it makes more sense that a larger wound volume will be "more effective and allow for more error" but who really knows.
I do know for a fact that there are a number of guys in Montana and across the US that are using 22 centerfires very successfully for bears, mule deer, whitetails, hogs, etc. 223, 223AI, 22 250, 220 Swift, etc with the super bullets are pretty effective. On 24hourcampfire.com there are hundreds of posts on 22 centerfire use on big game and the results are always very positive.
I generally hunt with my 308 or a 30-06 but I would have no qualms using a 22 centerfire for deer here in NB at any sane range (with for me is 300 yards max with any caliber) with a tough bullet (partition, tsx, 64 grain PP, 60 gr hornady sp). I base this on my penentration tests, emperical results on two animals, and a lot of information from the guys in the US.