Yes I had once.

Even as a teacher, some local companies need to provide all kind of papers to be allowed to hire a Western person and their funding capital also need to reach a certain amount. Otherwise, the company cannot hire through a Z visa, but only with a F visa for a short term business project.

Even if your "school" calls to ask what kind of documents they need to provide to hire you, it eventually ends up that you need to present more documents. The regulations provided for visa or resident permits, often state at the end that "Time Limit of Handling and Winding Up a Case: Within 5 working days, if application documents are complete."

But it's your school or the company who wants to hire you that must go through all the hassle and not you.

When applying for my first job in China, it felt like it took forever. For the first offer I applied to, it took them about a month after I turned in all the paperwork for them to tell me that I actually couldn't work there, because I didn't have the experience. But hey, after a week, they sent me another contract, this time with 2 F visas. That sounds cool, right? I had already rejected them after the first contract failed, I wasn't getting anything less than a Z visa. Only a few days after the first offer fell through, I found another contract with another school that sounded legit. That took 2 months after I first talked to them for them to send me the paperwork so I could apply for my Z visa. It was the most nerve-wracking summer I've ever had. Everything has gone well, but for real, waiting for it to go through is like torture.

Only 5 weeks? Sounds like everything is going well. Reminds me of some commercial where in the office, there are a bunch of monkeys playing around. You have to wonder what actually goes on and why the paperwork takes so long. It's a total lack of organization. There isn't really much you can do, just wait and keep insisting that you have the paperwork, resubmit documents if you have to. What are you going to do, sue them?

It was a hazzle for the first time but my company took care of everything, all I needed to do was just to wait at home.

The process of Z visa had been really hard, and will become harder, for the last couple of years, they are cracking down on illegal foreign workers.

I was recruited but a agent online, it took less then 2 weeks and they sent me a invitation, i got a 1 month visitor visa. They said its the best way yo come. i arrived and we completed all documents and again it as quick.
Today i understand why i had to come with a business visa, they lied to me about my employer, It wasn't a university (as they said) but a English center, but they treated me well, gave me a good apartment, paid me on time, i didn't do office duty and work for about 8 hours a week.
I don't have a TEFL, just a university degree.

That happened once to me. I guess some guy was on holiday or something, and the school (who was helping me with my visa) just got me a couple month-long F Visas while I waited.

As long as you have a Visa of some type, you should be OK in the interim. Although, I think you said you lived in Guangdong, and I hear they're a lot more strict about the rules there.