I always assumed unlocking your Tesla with the phone would use Bluetooth, NFC, even ad-hoc but I never expected it to be dependent on a round-trip to a internet-connected server. That seems like a really crap design and ripe for security issues, especially since it seems the Tesla customer support team has the ability to unlock cars remotely too.
The first commenter that says they can't enter their car also said their keyfob died that morning. Bluetooth keys work but not all Tesla cars have those. Some cars unlock by phone via the app through an internet connection, it seems. I'm surprised remote unlocking is as easy as calling a number and convincing an operator that you're the owner. Another reason to avoid the brand, I suppose.
I've consulted on on stolen luxury vehicle cases and I can tell you this isn't unique to Tesla. If you know what to say and can say it convincingly you can locate, disable, and unlock a higher end car pretty easily.