Yes, F visa too.

Info on Chinese visas from Chinese Embassy in Hong Kong:

http://www.fmcoprc.gov.hk/eng/zgqz/bgfwxx/

Why shouldn't they? There are competent English teachers from non-native speaking countries too. Well educated in the field of English, Education and alike, with clear pronunciation. And no, I'm not a teacher, but it doesn't seem fair, that some non educated drunk from native speaking country who didn't have any other option should get a head start over somebody who devoted himself to teaching for real.

That being said, this is mostly not the case, but shouldn't put everyone in the same basket.

I knew people from the Phillipines...Kongo..Zimbabwe etc., just to name a few places that had either an F or Z visa...

Why does it matter? One of the best ESL teachers I knew in my time here hailed from Romania. I've also had the pleasure of working with plenty of native fratboys/girls who slept through their own classes, passed out in their soiled bedsheets after a good night of pre-workday partying. Sometimes, the slightly more professional ones would come to work wasted, and then bumble home for a nap halfway through the workday. I can't emphasize enough how much training centers suck...especially if you are not, in fact, a drunken mess of a human being.

Native capabilities are barely a fraction of what it takes to do the job, my friend. If a non-native speaker speaks English well enough, I'd want to work with him more than my idiot native speaking (or slurring?) ex-coworkers any day of the week. At least I won't have their classes dropped on me at the last possible second, when they should already have been in session, because of their sloppy no-call, no-show behavior.

Maybe some places are beginning to realise, qualifications are not everything, and experience and ability count for something. Although, in fairness, mostly we are not wanted for our English ability as much as for the colour if our skins

Of course we can, I'm getting an X visa this week. 

Why wouldn't they give me a visa? 

I'm really going to study, with Chinese classmates and all. 

Just to be pedantic (and I think this thread deserves it), everyone is a native speaker of some language! And, I'd be fairly certain that those native speakers would be able to get F visas for some business arrangement or another.

Or, is the OP referring to NNS of Putonghua?? In which case - OMG!!!! Non-Chinese speaking business people coming to China! Oh, the horror!!!!

Edit: Damn, I missed it! What I should have said was "How does a country get a visa???

Competent non-natives should certainly be able to get visas, I work at the moment with some great non-native English speakers.

Ive met plenty of on native English speakers that speak better English than Native English speakers (besides me smart ass)    Maybe they could just go for SEX IN CHINA ON A 30 DAY VISA?