Yep. we do this too. Whenever we develop a web service for a customer, we set up uptime monitoring on the server where the app is hosted, but I also set up ping uptime checks on most of their other server infrastructure as well.
Just last week, I was alerted at 5am that a client's email server had gone offline. Nothing to do with us, we don't look after their emails at all, but I immediately texted their IT guy to let him know.
He texted me back at 6am - their own uptime monitors somehow failed to detect the DNS issue. He managed to fix it before their staff got to work that morning, and was extremely grateful that I had given him advanced heads up on something that was totally outside the scope of what they had engaged us for.
Context: We've worked with this client for 10+ years, I know their IT team really well. They are not a super huge organisation, and we are a tiny (read: 3 man) operation too, so the reliance on each other's services is something we consider important to keep everything moving forward.
Just last week, I was alerted at 5am that a client's email server had gone offline. Nothing to do with us, we don't look after their emails at all, but I immediately texted their IT guy to let him know.
He texted me back at 6am - their own uptime monitors somehow failed to detect the DNS issue. He managed to fix it before their staff got to work that morning, and was extremely grateful that I had given him advanced heads up on something that was totally outside the scope of what they had engaged us for.