Thanks again for all who responded. I am done with my trip and have the following notes:

I have a USA Chase bank debit card.

There aren't many ATMs anymore at PVG. I found Communication Bank, ICBC bank ATMs. There may be more, but I got tired of walking around.

ICBC and Bank of China seem to consistently allow $3000 cash withdrawal. I had 2 transactions of each in Shanghai and other cities (Zhangjiajie, Liuzhou) The limit may not be hard. As the above poster noted, the (paper) posted limits do not apply to everyone, and probably best-case-scenario for people with account at that particular bank. I figured out my limit ($3000rmb per transaction) by trial and error.

Construction bank and Communication Bank gives out $2500 at most. They tell you the limit when you get to the menu.

Postal Bank of China doesn't seem to let me take out money ("Invalid card"), although it let me enter my pin and go all the way to the withdrawal amount.
ChinaTrust (China CITIC Bank). 中信银行 - I tried to find 2 of these ATMs in Shanghai and both times, told me I had insufficient fund, even if I tried to take out $3000. It did give the option of taking out $5000. I guess if I had the right bank card, it would work... I know my card works because I literally walked over to Bank-of-China and took out $3000 right away.

One thing I noticed is that I always look for those international cash withdrawal symbols at ATMs to make sure the network is compatible with my banks (CIRRUS/PLUS network). However, my experience was that at many of these Chinese ATMs, even though I didn't find the right logos posted, as long as the ATM gave me the choice of accessing my savings/checking account, it was a proper ATM withdrawal and NOT a credit card cash advance.

Anyway, as Alipay and Wechat pay are prevalent for the Chinese, and starting to work for foreigners, this becomes less relevant. On this trip, I traveled to Guilin, Zhangjiajie, Liuzhou, and Siguliangshan (4 sister mountain? in Sichuan). Even out in the country side, taking local bus/vans, they always have the 2-D barcode for you to pay. Most vendors seem annoyed that I'm paying with those annoying paper money, but took it anyway.