Some of them definitely can jump - there was a lady in Leicestershire who used to hunt hers regularly.
The mare we had on loan before had jumped with her owner, and I show jumped her over small fences. I even took her hunting once, and while all the galloping about blew her brains somewhat (we have a photo of her with all four feet off the ground when I was trying to get her to trot down the road after the other horses

) she happily jumped cross country fences, even tho she had never done them before.
However, she would only jump if she felt like it, and woe betide you if you moved in the saddle on the approach.
If you're horse has done a bit of English, and will go forward from your leg into an 'english' contact, there's no reason not to have a go. But I would enlist the help of a sympathetic instructor - I once frightened the BSJA's head coach (a friend of our YM) to death when the mare jumped a fence and then did a sliding stop about 20yds after it - well she did ask me to stop