Everyone READ THIS PLEASE.
Here's what's going on right now: something with the PS3's internal clock has borked in some pre-slim models. This is not a PSN issue; it has to do with your PS3's clock. This clock is NOT related to the OS clock. It seems to be a leap year bug. Setting your clock ahead will do nothing. All PS3s around the world were affected at the exact same time.
Does this affect me?
-If you have a Slim PS3, you're good to go. Continue playing it as normal.
-If you have a "fat" PS3, you may be affected. Some fat PS3s have worked ok, but it seems a good majority have been bugged.
What does this bug mean?
-If affected, your date and time has been set to a random past date. This prevents you from signing into PSN, playing any PSN games, or using any DLC. Your trophies will also not display correctly or be able to sync.
How do I fix it?
-You can't. Even if you don't sign into PSN, it won't help. This is a hardware issue.
So, what do I do?
-Wait 24 hours. Other products that have had similar bugs have corrected themselves after the "bugged" day. On Monday around 8pm central time, all PS3s will automatically correct themselves...maybe.
-If that doesn't work, we'll need a firmware update from Sony, which should be a pretty important priority for them right now.
Did I lose my data/themes/etc?
Your save data/games/dlc/themes are fine. They'll be fixed when the PS3 is fixed. Trophies are uncertain right now, but they'll be fine if you've synced them before the PS3 was bugged. Unsynced trophies are an unknown right now.
If I have a bugged PS3, can I still use it?
It's probably not a good idea, as I have no idea what will happen to the trophies you earn during the bugged time. Movies/media will work fine.
So there ya go, my handy FAQ.