Just got one in my shop tonight. Here is where I am with it:
For me, the soft SMC reset did not work (Shift-Control-Option+Power button). So I reset the SMC by removing the back panel cover and unplugging the battery. With the battery unplugged from the main board, I connected the power adapter and it started right up. I didn't even have to hit the power button, which I believe is the default behavior after a hard SMC reset.
While still running, I then reconnected the battery. It showed up in the top bar as charged and so I unplugged the AC adapter and sure enough the MBA stayed running (battery works).
However, after shutdown - with the battery plugged back in - it once again will not start. So I pulled AC adapter, disconnected the battery connector from the board once again, waited a few seconds then plugged the AC adapter back in. It fired right up.
So at least in my case I can conclude that there is definitely an issue with either the battery itself or possibly one of the electronic components (capacitor/inverter/I'm not an electrician/). Will back the pictures up for the client and recommend sending it back to Apple for official battery replacement. Or she can risk something on eBAy or live with no battery.
Unit is a Mid 2011 MBA 1.7Ghz i5/4GB/128GB
Scrap the battery being the cause. Turns out there is a faulty power button. When I went to shut things down, I decided to use the button - nada. Even holding it down didn't do anything.
So I did a shutdown via the Apple menu. Unplugged the AC and the battery once again, and this time plugged in the battery first, then the AC. Presto - it fired up. However, I did them in a shorter period of time this time, less then about 5 seconds. The unit powered up right when the green light on the AC came on.
Forget the "default behavior after hard SMC rest" I posted above. Definitely recommending it go back to Apple.