I have a hopper minecart on a track that runs for weeks, then suddenly stops as if there is a block in its path. The hopper minecart cannot be found, and when another is placed the behavior observed in the video occurs. When I quit the game (this game is open to lan), and restart (sometimes after several quits and restarts), the hopper minecart will reappear but not in the block area where the problem occurred.
After re-entering the game after filming the video below, the missing hopper minecart was sitting on the track left and behind of the section I displayed in the video, and I could again send a hopper minecart through that space.
This has happened three times on different tracks and I do not know how to reliably reproduce the effect, but I have loaded a video of the glitch, and what I did to try to bypass along with the region file.
http://racinante.com/Minecraft/MC%20Glitch1.mp4
http://racinante.com/Minecraft/r.-1.0.mca
http://racinante.com/Minecraft/level.dat
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I have added the level.dat file but do not know if that will help. If the area self-repaired on exit, then the glitch will not show up in the region file. Windows would not allow me to copy the file with Minecraft running. If it happens again, and I think about it, I will use task manager to kill the program in hopes that the error will exist in the region file.
I found the area you showed in the video, however I was unable to reproduce the bug. It would be best if you can attach the whole world save here.

I found an earlier version of the game with the same problem. I loaded to:
http://racinante.com/Minecraft/TerraNovaTest.zip
Check the screenshots image for the location.
You will appear in the horse pen, open to lan, switch to survival and leave the pen and head south to the sugarcane farm. Into the hole with the chest, you will need to dig to the right of the test and down one block to access the rail system. Three-quarters of the way down (poorly lit, I know) there is a minecart stopped on the track. If you break that and collect the hopper and minecart, the detector rail will still registered as activated. You can then recreate the video by setting up the rail segment a couple of blocks to the right and left. You may also note that there are no hit boxes on the track (other than what you can see visually).
I will note that we do occasionally experience quite a bit of lag which may contribute to the problem.
If you save the game and exit and reenter, the invisible minecart will reappear.
Can reproduce, however this is most likely a dupe of MC-65040. After relogging, a second minecart appeared on the detector rail.
Cannot reproduce with given region file.