Arduino with wifi shield and 5AA batteries current draw
As a general answer, yes. AA alkaline should be more around 2500~3000 mAh depending on quality. 5x AA in series would result in 7.5 Volts and 2500 mAh.

But that's only for almost fully charged batteries. AA batteries can range from 1.6V to 0.8V "dead for most purposes". As they drain, the voltage drops. 

What may be happening is that you are powering the Arduino through its power jack, which goes through a diode, causing a 0.7V drop. Then it goes to the linear regulator which has a very high drop out voltage. Or even via the VIN pin, bypassing the diode. But as the battery drains, the voltage drops below the regulator ' s limit.

The Arduino microcontroller can easily work below 5V. The shield might not.