I phoned T-Mobile tech support and they got my problem sorted out. I'm now reporting back for the benefit of anybody in the future who may happen to Google "Currently unable to download. Please try again later" and wind up looking at this thread.
From the home screen, click on menu/settings/more/mobile networks, then click on "Access Point Names." Once you're on the Access Point Names screen, hit your menu button and choose "Reset to default."
Then get a friend to send you an mms message with a picture in it to test whether the problem is fixed. That's what fixed it for me, anyway.