I've seen this for several releases (also see bug 659277), but this precise report is with 3.6.2 on Fedora 18. My logs are full of garbage so it's hard to tell, but maybe this is interesting (I know this was mentioned in another report), and it's repeated many times in several occasions:
      JS LOG: Removing an access point that was never added
      JS LOG: Removing an access point that was never added
      JS LOG: Removing an access point that was never added
      JS LOG: Removing an access point that was never added
      JS LOG: Removing an access point that was never added

(gnome-shell:1809): Clutter-WARNING **: clutter_actor_raise Actor '<unnamed>[<Cl
utterGroup>:0x903a400]' is not in the same container as actor '<unnamed>[<Clutte
rGroup>:0xbffc510]'
      JS LOG: Removing an access point that was never added

The Clutter error does not always appear with the other message, but maybe it appears when things start going wrong, not sure. 


Note that once the bug happens, the connecting icon does not go away, at least
for some time, even when disconnecting/reconnecting, or even using a cable
connection. What's even weirder is that right now I'm on cable, and the icon is
a connecting WiFi; if I turn on the WiFi, the icon becomes a connecting cable;
if I turn it off again, the icon goes back to a connecting WiFi.
Disconnecting/reconnecting the cable has no effect on the icon. The only way to
get the correct icon to be shown is to disable both connections at the same
time. Maybe that's a hint.

This morning, after resuming from suspend, the indicator was still stuck in connecting WiFi state while I was on WiFi; turning off WiFi made the network indicator use the connected cable icon, and then the connected WiFi.