Since much of this lives in the PlayStation support, but not quite as handy for Home users as they might be, let me present my all-inclusive, super-awesome helpful post. I'll keep this updated and add to it as more presents itself.
The first place to seek help with connection issues is to check out this link:
How do I connect the PlayStation 3 computer entertainment system to the Internet?
(Be sure to check out the interactive section near the bottom titled "Configure networking settings")
You'll also want to check to see if the PlayStation®Network is down.
If the PlayStation®Network is down, there will be a real time announcement on the Knowledgebase Support Page.
Weekly Home Maintenance
PlayStation®Home has a weekly scheduled maintenance every Wednesday from approximately 1:00AM PST to 8:00AM PST.
Have a more complex set up, including a firewall (hardware or software), router in the house and/or multiple PS3s? You may need to adjust some of your ports, IPs, and permissions.
What are the firewall (TCP and UDP) port numbers to connect online with PlayStation® products?
In additional to the PSN ports listed in the above link, Home has a few additional ports and IPs required for full functionality:
TCP: 80, 443, 3478, 3479, 3480, 5223, 8080
UDP: 3478, 3479
Hopefully this will shed enough light on your topic to get things working. For all other connectivity issues, next try to search for further help in the Knowledge Center, then if you still have any issues please contact Customer Support for further assistance.