We are sorry to hear about the issue.
Could you please first copy the video to a folder on your desktop and then import them in Lightroom rather than directly importing them in Lightroom from your phone.
Please share the result with us.
I am having the same issue with importing from my Canon PowerShot SX700HS and iPhone6S. I have copied the MP4 and MOV files to the desktop as indicated and that did not make a change. I am running Lightroom 2015.2.1 under Mac OS X 10.11.1 (El Capitan update 1).
Is there an issue with the videos being 60fps?
Same issue with me when importing MP4 files from folder on primary (C) drive.
MP4s were created by phones of various manufacturers including Motorola and Samsung.
"Some import operations were not performed.... The video file appears to be corrupt or unrecognizable. It has no audio or video streams. (5)"
This is untrue, however, as the videos can be viewed perfectly and without any issues.
Lightroom version: 6.1.1
Operating system: Windows 7 Business Edition
Version: 6.1 
Application architecture: x64
System architecture: x64
Logical processor count: 8
Processor speed: 3.3 GHz
Built-in memory: 12270.4 MB
Real memory available to Lightroom: 12270.4 MB
Real memory used by Lightroom: 1426.0 MB (11.6%)
Virtual memory used by Lightroom: 1365.3 MB
Memory cache size: 2224.7 MB
Maximum thread count used by Camera Raw: 4
Camera Raw SIMD optimization: SSE2
System DPI setting: 96 DPI
Desktop composition enabled: No
Displays: 1) 1920x1080, 2) 1680x1050
Input types: Multitouch: No, Integrated touch: No, Integrated pen: No, External touch: No, External pen: No, Keyboard: No
Graphics Processor Info:
Check OpenGL support: Passed
Vendor: NVIDIA Corporation
Version: 3.3.0 NVIDIA 354.42
Renderer: Quadro 600/PCIe/SSE2
LanguageVersion: 3.30 NVIDIA via Cg compiler
Could you please update Lightroom to 6.3 and then try to import videos.
If that does not help, please import the videos first to your desktop and then try to import in Lightroom.
Thanks for your time and input Sarika.
1. I'm reluctant to upgrade to 6.3 because of the well publicized post 6.1 LR rollback and problems. Most everything is working fine, and I don't want to introduce any errors into my workflow or data.
2. I tried placing the videos on the desktop, but no joy. Windows Media Player opens and plays the problematic MP4s perfectly. Perhaps it is must be something in the MP4 wrapper as stored by the phone manufacturer software.
I'm experiencing the same error from .mp4 and .mov files from an Android phone, Sony compact digital camera, and Canon dSLR. Trying to import the files directly from memory cards or directly from my hard drive do not work.
I'm using Lightroom 6.7 and Camera Raw 9.7.
I am having the same problem:
1. Importing from the hard drive: "There is a problem with the video file (1) C0003.MP4"
2. Importing from an SD card: "An unknown error has occurred while reading the video file. Connecting to Dynamic Link server failed. (1)DSCN5681.MOV"
3. Both movies play okay on my PC.
4. Installed Lightroom CC a week ago onto Windows 10, that is up-to-date. Have never been able to import any videos. Lightroom 2015 Release 8. Everything else seems okay.