There is no hard limit to how many YN-560III flashes you can set off remotely with the appropriate transmitter device, as long as the flashes can receive the radio signal. Four is well within its capabilities.
However, the YN-560III only has a built-in radio receiver. You still need a radio transmitter. The following Yongnuo models can all be used as a radio transmitter on the camera hotshoe for the YN-560III and YN-560IV:
- YN-560-TX: dedicated transmitter unit, gives the most control over the remote flashes with six groups and manual power and zoom settings--but only over the YN-560III.
- YN-560IV: speedlight unit with radio master, only three groups can be controlled.
- RF-602 transmitter
- RF-603 transceiver
- RF-603II transceiver
- RF-605 transceiver; allows for group on/off control as a receiver on a non-YN-560III, but obviously no power or zoom control.
However. Your problem with the Neewer TT520 probably has nothing to do with the triggering system and everything to do with your not giving the flash sufficient time to recycle between bursts. The flash's capacitor has to gather enough charge from the AA batteries in it, to create a flash burst. The higher your power setting, the longer you have to wait for the recycle, which is why these flashes come with the recycle "beep" feature to let you know when it's ready to fire again. This is normal. If you are burst shooting, or using MULTI mode, then you probably need to be at very low power settings (1/32 or lower).
Your D7000 does have a commander unit in the pop-up flash, but this is not a radio commander. It's an optical "smart" commander for Nikon's Creative Light System (CLS). This protocol is light-based, and uses flash bursts, kind of like Morse code, to communicate the full hotshoe protocol so features like iTTL and FP (high-speed sync) flash are possible. They're not with the YN-560III--that's why it's so cheap. To use your built-in commander, you need a flash with a CLS slave sensor in it, like a Nikon SB-700, or a Yongnuo YN model with a name that ends in "EX" (or other CLS-slave capable 3rd party flashes, like those from Sigma, Nissin, Metz, etc. etc.)
Understand that this is also a completely different (and incompatible) optical triggering system from "dumb" optical triggers, such as Nikon's SU-4 mode, or the Yongnuo S1/S2 modes. In these cases, a simple "see flash burst=>fire flash" signal is all that's possible, and all the Morse code signalling from a TTL optical system will set this type of slave off early. To use this, you'd put a Nikon flash in SU-4 mode, or a Yongnuo flash in S1 mode, and then put the camera pop-up flash into Manual mode, and out of any TTL wireless scheme (master/commander, slave, wireless, etc.)