I've had the same issue with my BlackBerry, but I seemed to find a work around (although it's not very elegant).

Rather than signing in using your SkyMiles number, pull up your itinerary by typing in your name and record locator (this can be found on the bottom half of the "Check-in" page). The site will then successfully pull up the reservation and you can check-in the way it was meant to be

Since I get the "It's time to check-in" emails on my BlackBerry 24 hours before the flight, I just copy the record locator from that email and paste it into the browser. It's not ideal, but it works.

Hope that helps.