[aiharkness]

BBM definitely will (or "can" may be the better word, not sure) on wifi alone. Same for all the BlackBerry apps. If you lose connection to the RIM network, however, then no, even if you are still connected to wifi. (If the white portion of the wifi indicator changes from bright white to dull/grey, then you are not connected to the RIM/BlackBerry network.)

First, have you done a hard reset? If not, with the power on, remove the battery for at least 30 seconds. Then try again. Pay attention to the wifi status indicator.

If it is your router, also try a full power off/on cycle on the router and modem.