Just to get some more details, what's your router? And I take it "invisible mode" just turns off the SSID broadcast, and that's it?
Have you tried manually connecting to your router when it's in invisible mode? Just taking a shot in the dark here, since the iPhone initially connects when it's visible, it may think the router has dropped off entirely. Whereas if it knows it's connecting to a router that doesn't broadcast its SSID, it might behave a bit differently.
I had the same problem with an iPhone 4 (on a previous version of iOS) and a third-party router. My solution was to turn off invisible mode.
Did you by any chance recently upgrade the version of your phone?
