I get email on my phone - I run IMAP, so it is a duplicate of the inbox on my home PC - and the phone service-provider uses a different network to my terrestrial ISP. So for me, it makes sense.My internet was out for about five minutes. As soon as it comes back on, I get an email telling me there’s a service outage in my area. *shrug*
I also manage remote networks, so it is quite useful to get an email from the local (to the remote equipment) ISP telling me of an outage. While my own end-to-end monitoring will tell me of the lack of end-to-end connectivity, it's nice when the ISP owns up to the outage being their 'fault' so I don't have to start fault-finding the whole end-to-end service.