Do this test: make sure you are on a Wifi network with decent bandwidth to the Internet.
Send a high-quality video through that connection. It should work.
If it does, then it simply means that your phone can handle taking and sending good videos--it's the 3G or 3.5G or 4G connection you have that can't handle the video size and is pooping out before the upload finishes.
If that's the case, you might have a cellular antenna problem. Some Retina iPads had this, as well as various versions of iPhone 4 and Android phones. Or it could simply be that your cellular provider has crappy signal where you happen to be sending your hi-fi videos.