I'm a computer engineer and should be able to diagnose the repair this problem, but iOS is new to me...
My iPad2 (iOS 5.1) is configured for both my home and work WiFi. (the subject line is home config).
Safari looks to work fine at home and at work. So I have basic connectivity. BUT email and other Apps work fine at work but DO NOT have network connectivity when at home. I have iCloud, Gmail, and Verizon email access configured. NONE of them connect at home, but all work fine at the office (so it's NOT a particular email client setup issue).
Tonight I'm going to use Wireshark in an attempt to diagnose what is on the WiFi and/or wired LAN. I looks like DNS might not be working (I'll try explicit IP's). But DNS works ok on HTTP connections.
Other wired and wireless machines around the house have no problems accessing the same email services. But like I said, I have limited iOS knowledge.