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Am not sure about the latest versions of instagram, but older version you are only able to share existing video that must be recorded with instagram application. I found this link (http://www.youtube.com/watch?v=IoLCa-HBUG4), this might answer your querry. You may also find plenty of Instagram clone apps to share your existing videos.
In its 4.1 update, which rolled out this week, Instagram added the ability to import videos from your Camera Roll to the app. Read more here: http://www.itworld.com/unified-communications/368263/instagram-update-adds-video-imports-long-last
Hey I found this solution to get it to work on iOS at this link with a video:
Now you can find some ways to upload pre-recorded videos from your iPhone/Android phons to Instagram by googling.
Moreover, I made an Android app using the ways. InstaVideoUploader (:
In case of iOS, till now it is impossible to make the iOS uploader app because accessing other app's folder is not allowed in iOS. Howoever you can upload videos using iExplorer in iOS devices.
Yes you can, but it's not easy or supported by the instagram app. You need to interupt the video upload and replace it with a video rendered to the same size and specs. More detail here: http://techenvy.com/hardware/iphone/uploading-external-video-files-to-instagram