I am not a teacher, but I am an experienced student, and I can't understand teachers who don't have some backup solution: spare headphones, WiFi and mobile internet, laptop and tablet/smartphone. Once I had a trial lesson with a teacher with bad internet. Guess if we had the second lesson?
About the internet, be ready to use mobile internet in case of a problem with wifi, have an alternative to Skype like Whatsapp/Viber/Hangout. Switch off video. In the bad case reschedule the lesson.