Whenever I watch videos through the YouTube app on my iPhone 4 (iOS 6.1.3), after being on wireless for a while it will often spontaneously disconnect and start downloading from the 3G network. I only have a 200 MB data plan so this is a major problem. I'm guessing it is not app-specific, I just only happen to notice it with YouTube. I have 3/3 bars of wireless signal and my laptop in the same location has no problem so I don't think it's an issue with the wireless. I've tried restoring various levels of default settings and the entire phone to no avail.
I wish I could comment, but do not have enough reputation. Since it sounds like this problem is isolated solely to your specific iPhone, it sounds like it could be a problem with your router, or maybe the antenna in the phone itself is some how damaged. I know you said you don't think it could be a problem with your wireless router, but do you experience any of the same problems when using any other devices? My recommendation would be to reset your router to factory settings and then using your iPhone to view or download a large file over that connection.
I really do think it is a problem with your router, as I have seen other wireless connections exhibit the same problems when they are about to fail.