Okay, I may have figured out the battery issue, but it still has me scratching my head.
On the Adafruit webpage for the battery (http://www.adafruit.com/products/962
), the description clearly covers the fact that the battery has two USB ports, a 500 ma port and a 1A port. The description also clearly states that the 1A port is the one that should be used, because the the Raspberry Pi model B needs at least 700ma to run, and probably even more if you are using a WiFi stick like I am ... On the battery, the USB ports are labelled "Output 1, Fast charge" and "Output 2, Standard charge", with output 1 being the one closest to the power switch and LEDs. The label on the back also states that Output 1 is 1A and Output 2 is 500ma.
During my initial testing I was using "Output 1" (the 1A port) as suggested in the Adafruit documentation. When that consistently failed for me, I looked at the pictures provided on the product page and note that both of the pictures showing a Pi plugged into the battery, had the Pi plugged in the location of where "Output 2" (the supposedly 500ma port) is on my battery - the USB port farthest from the power switch and LEDs. When I tried booting using Output 2, the system appeared to boot fine (even with the WiFi stick in place). The system has been up and running now for over an hour without issue, whist when I was using the "Output 1" port, it failed to remain up for 5 minutes.
I was wondering if someone else with access to this battery can verify my finding. I am wondering if my battery ports are mislabeled, the documentation on the Adafruit site is wrong, or the Adafruit pictures of the battery are wrong.