Is it possible for a device to consume more power than it is wall-charging? I have a 2012 Nexus 7 running KitKat. I have a hunch that it is not charging properly. Specifically, if the system is on and the device is plugged into the USB wall-charger, the battery indicator says "Charging" but there is a slow decrease in charge over time. I have few apps actually running, some checks over Wi-Fi every 5 mins or so (email, feeds), and the device is unused most of the time. It charges to full if switched off and left plugged in.

I obviously need to investigate this more closely, check screen and app power consumption, etc, but before I go on a wild goose chase... is it actually possible for a device to consume more power than it receives via wall-charging?
Yes, it's definitely possible.  Most USB chargers are fairly limited in their output and devices like tablets are not really designed to be used while charging, unlike laptops.  You'd think it would be OK to simply sit and remain on while charging and unused, but perhaps the Nexus 7 does not have very good power management.  You might want to look at having sync pause while the device's screen is off.

If you want to crunch the numbers and verify what's happening, take a look at my answer here about the PS Vita.  If you get the specs/numbers for the Nexus 7 and its charger you could do a similar analysis to see whether the power use is exceeding the charger's output.