I was just able to finally reproduce the entire scenario. This not only can effect borrowing but can shutdown the License Manager as it eventually did in my test machine. Here is what I found.
Setup:
OS: Win7
Product: LM 10.4.1
Synopsis: Could not borrow licenses at first from the LM then could not connect to LM, eventually the LM stopped running. Tried to restart the LM. (LM service indicated that it was running, License Administrator indicated that it was not.) Tried stopping and starting the LM. Tried stopping the LM service. Could not get the LM running again.
Conclusion:
WARNING: The following "fix" required going into the registry which can render your machine inoperable if the wrong item is deleted or moved! Contacting Esri Technical Support is recommended.
Ran ProcMon and discovered that several registry entries were missing. Tried a REPAIR, No luck. Tried uninstalling and reinstalling, No luck.
Finally had to completely remove all directories and folders, remove registry entries and then run an uninstall to get everything cleaned out. At this point I ran a clean install using the "Run As Administrator" (even though I am an admin on this machine) in order to get it to work.
I will be submitting a bug on this however be aware that this has be very elusive and very, very difficult to reproduce so it may take quite a while to track it down.
In the meantime I would suggest that you contact Esri Technical Support if you run into this issue. Especially since it will probably require going into the registry which cannot render your system inoperable if you delete/move the wrong thing!