Hi Andrew,
Netlify is running ATS(Apache Traffic Server).
Without going too deep into deployment methodology, we drain connections to existing servers and take them out of rotation before doing any planned operations that would affect visitors. Since our DNS TTL is 20 seconds, if you reuse a connection or the results of a DNS lookup for much longer than that, it can lead to downtime that only affects you, since “real visitors” will be routed appropriately based on their repeated DNS lookups.
I hope this helps, and please keep us posted on what you find.