Did you turn off 3G but turn on WiFi? Because here is what I've found.
I wanted to extend battery life while at home and I'd heard that 3G was worse on the battery than WiFi (may or may not be true). So, I toggled off 3G and turned on WiFi. Everything works just fine.
However, after the phone goes to sleep, approx 10 minutes later, WiFi is toggled off by Android (probably to save battery). At that point, you phone is not connected to 3G (you turned it off) and no longer connected to WiFi (it timed out). So, no internet activity until you wake the phone back up. Text messages & phone calls are fine because turning off the 3G does NOT turn off phone calls or text messages.
I tested this by turning off 3G, turning on Wifi, waiting 10 minutes for WiFi to go to sleep, then sending an email from my PC. After an hour of waiting (and not waking up the phone), no email notification arrived.
YMMV, but this is what I've figured out so far.